Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\mathbf{C}$ indexed by the same index set $I$. Assume further that there exist coproducts $A$ and $B$ of $(A_i)_{i\in I}$ and of $(B_i)_{i\in I}$, respectively. If $f_i:A_i\to B_i$ are morphisms for all $i\in I$, then there exists a unique morphism $f:A\to B$, which is the coproduct of $(f_i)_{i\in I}$.

I know the following results. The morphism $f$ need not be monic, even if all $f_i$ are monic. If $\mathbf{C}$ is an abelian category and if all $f_i$ are epic, then $f$ is epic. Therefore, I have two related questions. All references are very welcome.

Question I.In general (where $\mathbf{C}$ need not be an abelian category), does $f$ have to be epic, provided that all $f_i$ are epic? If so, could you please provide a proof or a reference? If not, could you please provide a counterexample (for a finite $I$ and for an infinite $I$, if possible)?

Question II.If $\mathbf{C}$ is an abelian category and if all $f_i$ are monic, then does it follow that $f$ is monic? If so, could you please provide a proof or a reference? If not, could you please provide a counterexample (for a finite $I$ and for an infinite $I$, if possible)?