Dear categorists A PhD student of mine, Merce Llabres, and I we have got involved in proving some properties of pushouts on toposes, similar (but somehow dual) to those known for pullbacks. We are worried by the fact that perhaps somebody else has already proved many of the results we are interested in. So, before struggling to prove some probably well-known things, we would really appreciate some pointers to literature on the topic. For instance: Assume you have a family of pushouts in a topos Ai ---> Bi | | v f v A------> B (all squares have the same bottom arrow f) with vertical arrows monic, and assume the pullbacks of (Ai -->A)_i and (Bi---> B)_i exist, say A0 and B0, and consider the obvious square A0 ---> B0 | | v f v A------> B It is a pushout square when the family of squares is finite, and for arbitrary families in all complete toposes we have tried (sets, hypergraphs, total unary algebras, unary partial algebras with closed homomorphisms,...). Moreover, a proof (for complete toposes) can probably be derived from the techniques in the paper by Kawahara in TCS vol 77 (1990). But, has somebody already proved (or disproved) such a result? Thanks in advance Cesc Rossello