The second construction I gave in my last note ("more on inclusion maps") allowed a choice-free equivalence functor to one with an inclusion-map structure. A simple argument shows that one could use such not to find an inclusion structure on the original category (which we know is not always possible) but at least to find a choice nof monics, one in each set of names for a given subobject. And that of course would imply the axiom of choice (see below). Yikes. In case one needs a proof, given a family of sets construct a category as follows: for each S in the family let A_S B_S be names of objects. The home-sets of the form (B_S, B_S) each have only one element; S itself will be the hom-set (A_S, B_S), the complete group of permutations of S will be the home-set (A_S, A_S), all other hom-sets are empty. The composition of the endomorphism of A_S with the maps to B_S is just what you would expect. All elements of (A_S, B_S) name the same subject. Hence a choice of a monic from each set-of-subobject-names yields a choice function for the non-empty sets in the given family. The second construction doesn't work. The much easier first construction -- fortunately -- does work. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]