Dear Eduardo, Indeed I misunderstood what you wrote. My comments were on an arbitrary geometric morphism F -| U : EE -> SS. You, however, speak about a geometric morphism F -| U : SS -> EE where EE = Sh(CC,JJ) with the site (CC,JJ) living in SS. Then we know that F is determined by the restriction along Yoneda giving rise to a functor CC -> SS which you call F. Since CC is internal to SS we can speak about your F within SS. But you also mention X in EE. So we can't fully stay within the internal language of SS. But we may consider for X in EE the canonical map c_X = \coprod_{C:CC,f : hom(C,X)} ---> X which is epic for the reason given by Marta and me. Since F : EE -> SS is a left adjoint F(c_X) is epic, too. But we also have to exploit that F considered as a fibered functor is cocartesian, i.e. preserves internal sums. So you statement can be formulated in the internal language of SS when X is fixed externally (since EE over SS is locally small). But for proving it one has to go to the indexed/fibrational level. At least I don't see any other way. Best, Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]