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Two categories C and D are equivalent if and only if there are two functors F: C®D and G : D®C such that G ° F @ idC and F ° G @ idD (note that C is isomorphic to D iff G ° F = idC and F ° G = idD). 3 Proposition Let F,G : C® D be functors and t : F® G be a natural transformation from F to G. Assume that, for each aÎ ObC , ta Î D[F(a),G(a)] is an isomorphism. Then t is a natural isomorphism. 47 3. Functors and Natural Transformations Proof Define t-1 : G®F by t-1a = (ta)-1. t-1 is natural, since "fÎC[a,b] t-1b ° G(f) = tb-1 ° G(f) ° ta ° ta-1 = tb-1 ° tb ° F(f) ° ta-1 = G(f) ° t-1a.

1 Definition A coherent structure is a pair (|X|,- ), where |X| is a set and - is a binary, reflexive, symmetric relation on |X|. The elements of |X| are called points, and the relation - is called coherence. The coherent domain associated with (|X|,- ) is the collection X of subsets of P(|X|) whose points are pairwise coherent. The elements of X are ordered by set-inclusion. Coherence is extended to X in the obvious way, that is: A - B iff AÈBÎX. Exercise Prove, when X is a coherent domain, that 1.

Note that C´ ´ C[_,(a,b)] ° then < , > is a natural isomorphism from C´ D and C[_,a´b]: C®Set are contravariant functors. Conversely suppose to have for all objects a ´ C[_,(a,b)] ° D @ C[_,a´b]. The naturality and b an object a´b and a natural isomorphism t : C´ of t-1 is expressed by the following commutative diagram: 51 3. Functors and Natural Transformations Let (q1,q2) = t-1a´b(ida´b) . Note that q1: a´b®a, q2: a´b®b . We want to prove that a´b is a product with projections q1 and q2. Indeed, let f = tc(h,g): c®a´b in the above diagram, then we have (q1,q2) ° ( tc(h,g), tc(h,g) ) = t-1c(ida´b ° tc(h,g) ) = (h,g) and, in particular, q1 ° tc(h,g) = h ; q2 ° tc(h,g) = g .

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