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?
Let $C = 1$, let $A$ be an object such that $A \to 1$ is not a monomorphism, and let $B = 0$. Then, $A \times_C B = 0$, but $D = A$, so $D \to C$ is not a monomorphism. (For this to work we only need to know that $0$ is a strict initial object.)
Let us consider the topos of sheaves on the discrete space $\{ a, b \}$. Let $C = 1$, let $A$ be the subsheaf of $C$ such that $A_a = 1$ and $A_b = 0$, and let $B$ be a sheaf such that $B_a = 1$ and $B_b = 2$. Then, $f : A \to C$ is monic, $g : B \to C$ is epic but not monic, and both $p_1 : A \times_C B \to A$ and $p_2 : A \times_C B \to B$ are monic. But $D_a = 1$ and $D_b = 2$, so $D \to C$ is not monic.
Morally, what's happening here is that your hypotheses only guarantee that the restriction of $B$ (considered as a sheaf over $C$) to $A$ is monic, so you have no control over what $B$ looks like over the complement of $A$ in $C$. Nonetheless, this plays a role in the construction of $D$ and so influences whether $D \to C$ is monic or not.