I have a quick question ...
(Notation: $\mbox{Set}_A$ shall denote the category of presheaves on the category $A$.)
Suppose given two categories $A$ and $B$, and a functor $F \colon \, \mbox{Set}_A \rightarrow \mbox{Set}_B$ having the property that 1.) $F$ commutes with colimits, and 2.) $F_{|\,A}$, the restriction of $F$ to the presheaves that are in the image of the Yoneda embedding, preserves monomorphisms. Does $F$ then also preserve monomorphisms? Does it do so if the codomain of the monomorphism at hand is representable?
Thanks.