With reference to the following diagram in \CAT, where \set is the category of small sets: F \A------->\B | P| | v \set it is classical that if \A is small and \B is locally small then the left Kan extension of P along F exists, call it L, and for B in \B LB=\int^A \B(FA,B) x PA. Clearly, local smallness of B can be weakened to the requirement that all \B(FA,B) are small, a condition called `admissibility' of F, by Street and Walters in their `Yoneda structures' paper. The point is that a small integral of small sets is small. However, I claim that smallness of \A can be `weakened' to local smallness of \set^{\A\op}. If we include \set in a category \SET of sets large enough to contain all the LB as above, via I:\set----->\SET then the description of L also gives a left Kan extension of IP along F. Now write p(LB) for the power set of LB and consider the following calculation: p(LB)=\SET(\int^A \B(FA,B) x PA, 2)=\int_A\SET(\B(FA,B), 2^{PA}) =\int_A\set(\B(FA,B), 2^{PA}) =\set^{\A\op}(\B(F-,B),\set(P-,2)) It shows that local smallness of \set^{\A\op} implies p(LB) is small and hence that LB is small. Thus for \set^{\A\op} locally small and F admissible, the left Kan extension of any P:\A--->\set along F exists and is given by the usual formula. This calculation was prompted by questions similar to an open problem in the paper by Freyd and Street `On the size of categories' in TAC V1. Can anybody provide pointers to similar calculations where a big integral is tamed by a manageable end? Thanks, RJ Wood [For admin and other information see: http://www.mta.ca/~cat-dist/ ]