The other Peter J cites half of a theorem that appears on page 20, section 1.364, of Cats & Alligators. A "strong equivalence" was defined to be the usual notion: an isomorphism in the category-of-categories-and-natural-equivalence-classes-of-functors. The binary relation of "equivalent categories" can be taken just as the equivalence relation generated by the condition that there be a full embedding with a replete image. ("Full embedding" means only that the induced maps on hom-sets are isomorphisms.) If S is a skeletal category then it is a "skeleton of A" if appears as a full replete subcategory of A. It is "co-skeleton of A" iff there is an onto full embedding A --> S. Then: Each of the following is equivalent to the axiom of choice: (a) Equivalent categories are strongly equivalent. (b) Every category has a skeleton. (c) Every category has a co-skeleton. (d) Any two skeletons of a category are isomorphic. (e) Any two co-skeletons of a category are isomorphic. For convenience we add: (f) Given a non-empty family {S_i}_I of equinumerous sets there exists 0 in I and a family of isomorphisms of the permutation groups {Aut(S_0) --> Aut(S_i)}_I. (g) Given a family (S_i}_I of non-empty equinumerous sets, there exists a family (x_i}_I such that x_i in S_i for all i in I. [I've moved the string "non-empty" in (g) to its right place.] The first time I ever thought to write this down was in 1980 at a blackboard in one of those bomb-shelter-style classrooms in the old math building at Cambridge. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]