Suppose you are in the category of sets or more generally in a topos (i.e. sheaf topos) and $f:A\rightarrow C$, $g:B\rightarrow C$ are two morphisms.
There is a canonical map $u:D\to C$ from $D$ (defined as the pushout of the diagram $A\leftarrow A\times_C B\rightarrow B$ consisting of the two projections) into $C$.
Presumably $u$ doesn't have to be a monomorphism in general, however I can't think of a counterexample. In my situation, it is supplementary given that the two projections $pr_1:A\times_C B\rightarrow A$ and $pr_2:A\times_C B\rightarrow B$ and $f$ are monomorphisms each. Does $u$ have to be a monomorphism then?