CATEGORICAL CONSTRUCTIONS IN GRAPH THEORY

This paper presents some graph-theoretic questions from the viewioint of the portion of category theory which has become common knowledge. In particular, the reader is encouraged to consider whether there is only one natural category of graphs and how theories of directed graphs and undirected graphs are related.

combinatorics which leans towards defining graphs in terms of adjacency properties of a set of vertices, which restrict the available objects and ignores the relations between them, while category theory emphasizes the structure preserving mappings (morphisms) and gives its best results where one is able to construct objects "freely determined by some properties".
Our first observation is that directed graphs are easier to describe than undirected graphs.In Section 2, we construct categories which give satisfactory descriptions of a category of directed graphs and which have simple categorical descriptions (as functor categories).The construction of a category of undirected graphs is discussed in Section 3. Second, vertices and edges have traditionally been considered to be different things, but Ribenboim treated vertices as being degenerate edges.In Section 2,we study the relationship between the categories of directed graphs which model these two definitions.In Section 4, we observe that the "Cartesian" structure has more intuitively satisfying properties when the vertices are considered as degenerate edges.
Section 3 is devoted to recapturing an undirected graph from one of its canonically generated directed graphs.This process uses the theory of monads which stems from the definition of "algebraic structure" in the context of category theory.

TWO POSSIBLE DEFINITIONS OF "DIRECTED GRAPH"
In order to describe a directed graph G, one first specifies a set V of vertices and a set E of .Each edge is considered as starting at a vertex, called its origin and going to another vertex, called its terminal.Actually, these assignments define functions o (for origin) and t (for terminal) from E to V.This definition allows oriented graphs to have multiple (el,e 2 e E with e I e2, o(e I) o(e 2) and t(e I) t(e2)) and (e e E with o(e) t(e)).
Often, such graphs are excluded in combinatorlal graph theory problems.This definition is equivalent to defining a directed graph as a functor from the category A_ p (see Figure i) to the category Ens, of sets.If graphs are functors, then the appropriate definition of "graph morphism" should be a natural transformation between these functors.Such a natural transfor- mat ion f:Gl': G2:AP Ens is given by a map of vertices f[O]:Gl[O] G2[O] and a map of edges f[l]:Gl[l] G2[I] which "commute with" origin and terminal.Thus a category of directed graphs can be modeled by the functor category [A_P,Ens].
By contrast, considering vertices to be "degenerate edges" leads one to view the category of graphs as the functor category [BP,Ens] (see Figure 2).  said to be origin and terminal, respectively.In addition, the identities o0 i id[0], i 0, I in _B insure that the function H(o0):H[0] H[I] is an embedding of the set of vertices into the set of edges of H, mapping each vertex to a loop based at that vertex.Thus, each vertex can be thought of as a "degenerate edge".
Since [AP,Ens] and [BP,Ens] are both examples of functor categories, we are able to use general properties of functor categories to describe the fundamental category-theoretic constructions within each category, and to relate these two candidates for a category of directed graphs.The first principle which we use is that limits and colimits (including the terminal object as a special case of a limit and the initial object as a special case of a colimlt) are computed "pointwise".
For example, the null set is an initial object in Ens since there is a unique (null) map of into every set.An initial object in [AP,Ens] or [Bp,Ens] is a graph with no vertices, no edges, and all structural maps null.
The null graph is thus essential to a categorical approach.
Similarly, since a terminal object of ns is any one pointed set, the terminal object of [A__P,Ens] must have one vertex and one edge which is a loop.In     Since the same construction can product very different results in [A p,Ens] and [B p,fns], it is desirable to be able to relate these t-o categories so that constructions in one would be naturally "induced" in the other.
The inclusion u:A --B of A as a subcategory of B induces a (forgetful) functor U:[BP,ns] [AP,ns].
For a B-graph H, the A-graph U(H) is just the restriction of H to the sub- category A p of B_ p.In the A_-graph U(H), the degenerate edges H(o0)v, for various vertices v e H[0], are no longer distinguished.The restriction of natural transformations gives no difficulty, and it is easy to verify that U:[BP,ns] [AP,ns] is actually a functor.
Because U:[BP,Ens] [AP,Ens] is defined by composition with the inclusion op u :A p B p, it induces more structure: U has both left and right adjoints which are the Kan extensions L and R.These are described below.
Particularly, H carries a vertex v of the B=-graph H to the vertex of RU(H) which is the degenerate loop H(o0)(v), i.e.NH(V) H(o0)(v); and H maps an edge e of H with origin v and terminal w to the edge <H(g0)v,e,H(g0)w>, i.e. qH(e) <H(g0l)e,e,H(g060)e>. Clearly, H:H RU(H) is the B__-monomorphism which identifies H with the full subgraph of RU(H) generated by vertices {H(o0)vlv e H[0]}.
The counit e:UR'/Id:[AP,F-ns] [Ap,Ens] is analogously defined to be a family of A-graph morphlsms eG:UR(G) G.The vertices of UR(G) are the loops of G.The A_-morphlsm e G maps such a loop to its vertex G(O)() G(I)() and carries an edge <o,e,l > to the edge e; i.e. eG[O]() G(O)() and G[I]<0,e,I > e.Thus e G "compresses" UR(G) onto the full _A-subgraph of G generated by those vertices having loops.
Note that e G is an A-graph isomorphism if and only if at each vertex of G, there is precisely one loop.Similarly, H is a B_-graph isomorphism exactly when each loop of H is a degenerate loop H(o0)v.Therefore U and R induce an isomorphism between the full subcategorles A of A_-graphs with exactly one loop per vertex and 8 of B-graphs with only degenerate loops.
These ideas above illustrate the first part of the following theorem whose second part we use later.
THEOREM 2 1 (Lambek and Rattray) Let S:X V:T be a pair of adjoint functors with unit n:id -TS:X X and counit :ST *id:F V. Then T and S induce an equivalence between Fix(TS,) {X obXlx:X TS(X)} and Fix(ST,) { e ob,ly:ST(, )._2-y}.Moreover, the following statements are equivalent (i) the triple (TS,r,TS) on X_. is idempotent" (ii) T is a natural isomorphism; (iii) the cotriple (ST,e,ST) on is idempotent" (iv) _S is a natural isomorphism" (v) TS:X X factors through the subcategory Fix(TS,)" (vi) ST:-F factors through the subcategory Fix(ST,g).
If these conditions hold, Fix(TS,) is a reflective subcategory of _ with reflector the factorization (v) and Fix(ST,g) is a coreflectlve subcategory of F with coreflector the factorization (vi).
In the example just discussed, S:X_-F__ is U:[BP,Ens] [AP,Ens] and T:Fis the right adjoint R:[AP,Ens] [BP,Ens] The condition for Fix(RU,N) [BP,Ens] to be reflective and Fix(UR,E) [A p,Ens] to be coreflective fail.Consider RG:RG RU(RG) where G is a graph with one vertex v and two loops 0,%i at v. Then RG has two vertices {0,I}, each with two loops However, RU(RG) has four vertices, the four loops (2.1).Thus nRG fails to be an isomorphism and condition (ii) above does not hold.In fact, FIx(RU,N) is not coreflective since the inclusion does not preserve colimlts.
Suppose we compose the adjunction S:X__.._F:T with an adjoint pair -":N then Theorem 2.1 applies to the composite adjunction MS:XZ:NT.
M: Y _ _ _ In particular, if Z_ is a reflective subcategory of _F with idempotent cotriple (MSTN,g,MSTN), then there is a reflective subcategory X_l of _, guaranteed by Theorem 2.1 which is equivalent to a coreflective subcategory Z_l of Z_.
As an illustration, let F 1 be the full reflective subcategory of generated by all A=-graphs which have at most one loop at each vertex.The left adjoint (usually called the reflector) M:[AP,Ens]_ F I of the inclusion [A__P,Ens] maps each At-graph G to the quotient A-graph G 1 with the N FIsame vertices as G, the same edges between distinct vertices, but having only one loop at each vertex at which G has loops and having no new loops.Applying UR:[AP,ns] [AP,ns] to a graph NG 1 with GI I' yields the full subgraph UR(G I) of NG 1 generated by the vertices of NG 1 at which there is a loop.
Note that UR(G I) is already in the subcategory I; hence the function M:[AP,Ens] F I, has no further effect and MURN(G I) URN (GI)' NG I Iterating this construction yields the same graph, up to isomorphism.Thus condition (vi) of Theorem 2.1 holds, and all the other equivalent properties (1)-(v) and the conclusions of Theorem 2.1 follow.Particularly, the coreflective subcategory of F I consisting of A-graphs with exactly one loop per vertex is equivalent to the reflective subcategory A of B-graphs with only degenerate loops.Note that Fix(UR,g) and A Fix(RU,q) for the original adjunctlon [A p Ens] :R.Thus it is possible for Fix(TS,q) to be a reflective U: [B p ,Ens] subcategory of X_ without the existence of a factorizatlon of TS:X_/ X through Fix(TS,q) (i.e.condition (v) of Theorem 2.1 fails even though part of the conclu- sion is still true).
A second important subcategory 2 of [A_P,Ens] is the full reflective subcategory generated by the simple A_-graphs G with at most one edge from any particular vertex to another.The reflector M:[A p Ens] the left adJoint 2' of the full inclusion N:F2r--[A__UP,Ens],--maps each A__-graph G to the quotient A_-graph G 2 having the same vertices as G, but with edges e and e' identified whenever G(l)e G(l)e and G(O)e G(O)e '.Again, the cotriple on 2 is easily seen to be idempotent.Using an argument similar to the one for I' it follows that there exists a reflective subcategory A 2 of [Bp,Ens] which is equivalent to a coreflective subcategory 2 of Theorem 2.1 thus establishes equivalences between subcategorles of [_A_P,Ens] and [BP,Ens] which introduce the combinatorially useful concepts of "absence of (unnecessary) loops" or "characterization of edges by their origin and terminal vertices".The "combinatorially interesting graphs" are extracted in a natural way from either of the functor categories [AP,ns] or [BP,Ens].Thus either of the two possible general definitions of "directed graph" leads to the same theory.
Along the way, though, we have found that the interplay of these generalizations leads to deeper properties than one might have expected.

UND IRECTED GRAPHS
In combinatorial problems, a graph is simple considered to be one-dimensional complex.Each graph U has a set V(U) of vertice.s, a set E(U) of edges, and an incidence relation I(U) V(U) E(U) with each edge incident to at most two vertices.Clearly, a directed A_-graph G can be given A morphism f:U I U 2 of undirected graphs is given by a pair of functions, ) and E(f):E(U I) E(U2), such that incidence is preserved; i.e.
(V(f) E(f)I(U I) I(U2)- Any A__-graph morphism f:G I G 2 "induces" such a pair e(f) :e(G I) P(G 2) of functions between undirected graphs since commutes with origin and terminal.
Hence, this construction induces a functor P'[AP,Ens] T from the category of A_-graphs to the category T of undirected graphs.The category T is not easy to describe, since it depends on using, within category theory, the notation of a set with at most two elements (to describe the incidence relation I(U)).This difficulty illustrates a difference between the combinatorial and categorical viewpoints in graph theory.The categorical approach searches for general concepts which may be simply described and proceeds from there to particular or special examples" while the combinatorial approach assumes that such things as multiple edges or loops will be complicated.The "purity" of the categori- cal approach aside, it appears necessary to make some arbitrary choices in order to obtain a category to model T.However, we are able to avoid this difficulty by external relationships (which we expect to exist) between [P,Ens] and using to help "internally" construct T. Thus we search for objects OU in [P,Ens] such that they agree with our "intuition" about the category T and such that there is a natural equivalence (G,U):T(PG,U) /[A_P,ns] (G,OU). (3.i) Every A_-graph G is determined by the sets G[O] and G[I] of vertices and edges, respectively, and by the functions G( I) and G( O) of origin and terminal.
[P,Ens] is a functor category, the Yoneda Lemma guarantees that these Since sets and functions can be naturally described in terms of representable functors; in particular, G[ i] [AP,Ens] (A( ,s i) ,G) i 0,I.from V to E select the origin and terminal vertices, respectively, of the unique edge in m.The subcategory A' of [AP,Ens] consisting of objects V and E with morphlsms B and T is isomorphic to A.
Since graph V has one vertex and no edges, and P:[AP,ns] T is a forgetful functor, PV is the graph in T with one vertex and no edges.A T-morphism PV U is given by selecting a vertex of U to be the image of the unique vertex of PV.Thus (OU)[O] is identified with the set of vertices of U.
Similarly, the edge set of OU is given by the set T(PE,U).The A==-morphism T:V E induces a map T(PT,U):T(PE,U) T(PV,U).The undirected graph PE has one edge and two vertices; thus T(PT,U) restricts a morphism with domain PE to one of these vertices.If vertex v of U is incident to edge e, then there is a T-morphism f :PE/U such that the unique edge of PE is mapped to e and such that the vertex of PE designated by PT:PV PE is mapped to v.
We expect that this description above would determine the morphism f e uniquely.First, if e is a proper edge, then e has precisely one other vertex incident with it; and f :PE U should have this vertex as the image of the e second vertex of PE.Second, if e is a loop, then there is only one vertex v incident with e; and both vertices of PE must be mapped to v. Note that these assignments establish a natural equivalence between (OU)[I] T(PE,U) and I(U) V(U) x E(U).
In addition, the terminal map is given by the natural projection of I(U) on V(U).
To complete the description of the directed graph OU, it is necessary to give the origin map The argument given above to justify the identification of T(PE,U) with I(U) is a construction of (OU)(I); i.e.
(OU)(61)(v,e) Iv'; if (v',e) I(U), v' # v [ v; otherwise This construction uses the difficult to naturall[ describe notion "is incident with either one or two vertices".The existence of the adjolnt relationship <P,O> demands that the construction of the origin be natural.Hence we will exploit the simple functor category op Ens] of A-graphs use the assumed existence of an adjoint pair <P,O>, and apply the general theory of monads to construct a category of algebras F which will be a functor category model [yP,Ens] for T.
As a first step, we give a description of the monad (or .triple)M <T,,U> in [h p Ens] which results from the assumed existence of the adjunction <P O> of diagram (3.3), for each A_-graph G. Clearly, from the definition of TG( O) and TG(I)' G is an A-graph morphism natural in G.

T:[AP,ns] [AP,Ens]
of the monad M is constructed from the counit :PO id:T-T of the adjoint pair <P,O>.For each graph U, cu-PO(U) U is the identifying map on vertices, V( U) Id:V(PO(U)) V(U).where U is a "twist" map which chooses the other element of I(U) over the same edge in E(U).In the context of the monad M, the above constructions lead to the pushout description (3.6) for the multiplication NG:T2G TG: id where t: The axioms for a monad (see MacLane [2], VI.I) are now easily verified; in particular, the constructions are clearly "natural in G" G e [A p Ens] An alsebra for the monad M <T,,> is defined as an object X of the under- lying category, which is [Ap,Ens] in this case, together with a morphism h:TX X such that diagrams (3.8) ) and DX[I]:X[I TX [I] is the map 0 of (3.3), we construct h:TX X with hN X id x from the pushout (3.9): ld x []/zl____ b (3.9) The requirement that h:TX X be an A-graph morphism forces the map b: to reverse orientation (i.e., x(i)b X(6 l-i) :X[l] X[O] i 0,i) Furthermore the restriction b/Z(X) must be the "identity" inclusion (X) X Thus the concept of graph with involution arises naturally from the concept of unoriented graph; in fact, it provides an algebraic realization of this concept.The category T I of M-algebras always has a pair of adjolnt functors which induce the monad M in the base category.
In addition, the Kleisli construction uses the monad M to describe a category T 2 which is essentially the category of free algebras.The objects of T 2 are the same as those of [AP,Ens],_ but there are additional morphisms in T 2. Any pair of functions f <fo,fl >, fo: (rather than origin and terminal) is a morphism of T 2.
In general, the category of all algebras (in our case, T I) is a terminal object in the category Adj(M) of adjoint pairs which induce the monad M, and the Kleisli construction (in our case T 2) is initial in Adj(M).The unique :T 2 T I functor of MacLane [2] (VI.5, Theorem 3) is both full and faithful.Furthermore, every algebra in T I is isomorphic to a free algebra in (T2)" hence, the two categories are equivalent.Thus, T I and T 2 are, essentially, equally good models of T, and they are canonically chosen from all categories inducing the monad M.
The category T 2 is perhaps closer to our original idea of unoriented graphs but the category T I has other advantages.Observe that A__-graphs with involution can be naturally extended to a C_-graph (object in [c_P,Ens]) with C pictured in Fig. 5. [o]-.---.
T61 60 0 I 2 The category of algebras T I defined by the monad M is the full subcategory of [cP,ns] of C__-graphs U satisfying {elU(r)e e} {elU(60)e U(61)e} i.e., the set of edges fixed by U(T) is precisely the set of all loops of U. It is easy to show that T I is a reflective subcategory of [C_ p,ns].Construct the reflector, the left adjoint to the inclusion of T I into [cP,Ens],_ by passing from a given C_-graph U to a new C-graph UI, a quotient of U, having the same vertices as U, having the same edges between distinct vertices of U, but having U(r)e identified with e for each loop e (U).
Note that equivalence of categories is weaker than isomorphism.In particular, the comparison :T 2 T I is not an isomorphism of categories, i.e., the object map II:ITml TII is not one-to-one and onto.However, the notion of equiva- lence appears to be more appropriate than isomorphism for modeling a theory.
Hence, there is only one "useful" model up to equivalence, but two different approaches to that model.
We may also form a corresponding theory for B-graphs starting from an orientation-forgetful functor P':[B p Ens] T' T I The category of algebras for a monad M' in [BP,Ens], with similar proper- ties to the monad M in [AP,Ens], may be described by introducing an "involution".
These algebras approximate the "involutional graphs" of Ribenboim [5].Of course, if a graph admits more than one involution, each involution defines a different algebra.(The original definition required only the existence of an involution in the hope of defining a subcategory of [BP,Ens] but the involution was incorporated into the algebraic structure in Ribenboim [6].)Curiously, the definition in Ribenboim [5,6] also requires that an edge fixed by involution be degenerate.r'H(i):T'H[l] r'H[0], i 0,i, is given as in (3.4).Again, NH[I]:H[I] T'H [I] is the analog of z 0 of (3.3), and H[l] :T'T'H[I] T'H[I] is constructed as in (3.6) and (3.7).(Note that the mapping H' in general, is not as isomorphism (cf.Ribenboim [5], p. 159).) The category T I of all algebras for the monad M' in [BP,Ens]_ is the category [DP,ns], where D is depicted in Figure 6.The corresponding Kleisli construction yields a category T 2 which is not equivalent to TI; rather the free algebras in T 2 are characterized by (3.10).
Thus the original definition of involutorlal graph was designed to select a sub- category of T I equivalent to T 2. It is also curious that (3.10) defines a subcategory of T I [DJ p_ ns] which is both reflective and coreflective Also, consider the Kan extensions of the functors induced by the inclusions B-D and A''C.In particular, the forgetful functor U:[DP,ns] [BP,ns] and its left adjoint (i.e., left-Kan extension) induce the monad M.

CATEGORIES OF GRAPHS ARE CARTESIAN CLOSED
A benefit of viewing a category of graphs as a functor category is that any functor category [xP,Ens], for small X is cartesian closed (see Freyd [9], p. 8).
The proof of this result is constructive; i.e. an algorithm is provided for com-  Thus Hom(Y,Z)[I] is identified with the set of all triples of functions <r:Y The origin, Hom(Y,Z)(61), is given by (4.In general, the set of edges joining points s,t Hom(Y,Z)[0] is given by the set of functlo=s r:Y[l] Z[I] satisfying (4.2).Although this construction satisfies the adJoint property (4.1), it seems to have little relation to graph  3).The definition of Horn graph in Ribenboim [5], p. 165, was in this spirit.However, for no apparent reason, only the restriction of r to Y(0)y[0] entered into that definition.In Ribenboim [6], p. ii0, a totally different definition was given.Again, it was an explicit construction, and again there was no claim of naturality.The idea was that the edges should be indexed by the functions r:Y[l] Z[I] of our definition.The difficulty is that (4.3) determines only s[0] and t[0], so that the vertices could only be the equivalence classes of "graph homomorphlsms restricting to the same functions on vertices".By contrast, our construction is no more cumbersome than these two attempts, but the underly general principle is quite simple and guarantees the Horn will have the proper adjoint relation to cartesian product.
Our internal Hom-functor has the usual properties of Hom in ns in particular, there is a composition o:Hom(Y,Z) Hom(W,Y) Hom(W,Z). (4.4) The realizations of vertices in Hom(Y,Z)[0] as graph homomorphisms and of Hom(Y,Z)[I] as triples <r,s,t> are compatible with the composition (4.4).
The composition (4.4) induces a natural monoid structure on Hom(W,W).If the invertible elements of Hom(W,W) are selected, the resulting subgraph is a group called Aut(W).
The various reflective subcategorles constructed in Section 2 and Section 3 in the discussion of categories of graphs sould be cartesian closed.In fact, the following proposition gives an easy computation of Hom in many cases.
PROPOSITION.Suppose F is a cartesian closed category and I:F' is a full reflective inclusion with R:F F' the reflector (i.e., <R,I> adjolnt pair).Then R preserves products iff for all A F and B ', Hom(A,B) is in F' PROOF.See Freyd [9; p. 13].
REMARK.The reflective subcategory A I of [P,Ens]_ constructed in Section 2 does not satisfy this property; however A 2 does.

Figure 4 .
Figure 4.The B_-graph HH' this structure, denoted by P(G): the vertices are G[O]; the edges are G[I]; and the incidence relation is the collection of pairs {(G(60)e,e),(G(61)e,e)le e G[I]} G[O] x G[I].
The object (actually functor) V -= A=(_, O] :A p ns which represents vertices has one vertex since A([O],[O]) consists only of the identity, and no edges since A([I],[O]) is empty.The object E A(_,[I]):A p Ens which represents edges similarly has two vertices and one edge joining them.The morphisms (natural transformations) S-A( ,61):A(,[0]) i A(,[i])

For
each h-graph G, TG Of(G) is described using the above discussion.Hence, rG[O] G[O] and TG[I] is given as the pushout Zo ] IC( O) (e) G(I) (e) }, the se__t o__f loops of G.The universal properties of the pushout in ns are used to give a description of origin TG( I) and terminal TG( O) to reflect the above construction of T OP:[AP,Ens] [AP,Ens]. [hus TC(60)'IG[I] TG[O] is the composition of the identification G[O]'G[O] -TG[O] with the function in G( O) (t) while TG(I):TG[I] TG[O] is defined similarly by interchanging G( O) and C(6 I) in (3.4).The unit of the adjunction <P,O>, :id "-r:[Ap,ns] [AP,Ens], is given by rG[O] ld:G[O] TG[O], and rG[1 -= Zo:G[1 TG[1] The set E(PO(U)) of edges of PO(U) is, from the above discussion, just the incidence relation I(U) _V(U) E(U).A sketch of theconstruction of the equivalence (G,U) :T(PG,U)--[AP,[ns] (G,OU) leads to the function E(u):E(PO(U)) E(U) being the projection I(U) E(U).The incidence relation I(PO(U)) in PO(U) can be given as a pushout with a Note that t:TG[I] TG[I] together with id:G[O] G[O] TG[O] on verticesdoes not define an A-graph morphism, but G <Id'G[I]>:T2G TG is in [AP,Ens]._ [I].The commuting of the first diagram of (3.8) reduces, in this case, to the statement that b:X[l] X[I] has b 2 Id:X[l] X[I]" i.e., b is an involution.

Figure 5 .
Figure 5.The category C (3.10)The requirement(3.10) is incorporated naturally into the program developingthe monad M' by having H(oO):H[O]-H[I] play the role of the inclusion (G):' G[I] of (3.3) in the definition of T'H[I].With this change, T'H(o0) is the common composition H[O] T'H[I] in the analog of (3.3); and

Figure 6 .
Figure 6.The category D puting the "internal-Hom functors.If Y, Z are objects in [Xp,ns], then Hom(Y,Z) is also an object in [xP,Ens], whose evaluation at object p of X is defined by: Hom(Y,Z)(p) [xP,Ens] (X(_,p) x Y,Z).

Figure 7 .
Figure 7. Representable functors As an example of the use of formula (4.1), we construct Hom(Y,Z) in [AP,Ens].The set of vertices is given by Hom(Y,Z)[O]

2 )
the terminal, Hom(Y,Z)(60), by <r,s,t>-s <r,s,t>--t.EXAMPLE.If Y is discrete, i.e.Y[I] , then Hom(Y,Z) has all functions Y[O] Z[O] as vertices and has a unique edge joining each pair of vertices.
Each "edge" of Hom(Y,Z)[I] is represented by a triple <r:Y[l] Z[I]s:Y Z; t'Y Z>, where s and t are B_-graph morphisms for which (4.3) is commutative: (4.3)As before, <r,s,t>-s gives the origin of Hom(Y,Z) and <r,s,t>-t defines the terminal.If s and t are given, then the set of edges of Hom(Y,Z) with origin s and terminal t is indexed by those functions r:Y[l] Z[I] satisfying(4.
each A-graph G and each B-graph H.
To see that L: [AP,ns] [BP,ns] is the left adjoint of U:[B p ns] [A p Ens] one shows that there is a natural one-to-one correspondence (G,H):[BP,Ens](L(G),H) [AP,Ens](G,U(H)) for