Consider the category $mon(C)$ (objects as monics in $C$) as full subcategory of the $Arrow(C)$ . We know that $Arrow(C)$ is finitely complete and cocomplete, assuming that $C$ has limits as well as colimits. Does $mon(C)$ (finitely) cocomplete?
For example consider coproducts: For objects $m_1:A \rightarrow C$ and $m_2:B \rightarrow D$ in $Arrow(C)$, the coproduct is given by $[m_1, m_2]:A+B \rightarrow C+D$ (using the couniversal property of $A+B$). Does $m_1$, $m_2$ monic implies $[m_1,m_2]$ being monic?
I think it is not true in general. If it is assumed that $C$ is a topos, does that help in cocompleteness of $mon(C)$ in anyway? (For $C$ topos, $mon(C)$ will have exponents, but not subobject classifier).