# Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

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)?

Question 1: Yes. The $$I$$-coproduct-functor $$\bigsqcup_I\colon\prod_{i\in I}\mathbf{C}\to\mathbf{C}$$ is left-adjoint (its right adjoint is the diagonal functor $$\Delta_{\mathbf{C}}^I\colon \mathbf{C}\to\prod_{i\in I}\mathbf{C}$$), hence always preserves epimorphisms.
Question 2: No, in general (even if $$\mathbf{C}$$ is $$I$$-coproduct-complete). Cocomplete abelian categories with such property are called satisfying axiom AB4. See nlab article on Grothendieck axioms. Some counterexamples were discussed on SE; see, for example Zhen Lin's answer on MSE. The original source for this axiom is the Tôhoku paper (A.Grothendieck, "Sur quelques points d'algèbre homologique", 1957).
In the situation when $$\mathbf{C}$$ is not $$I$$-coproduct-complete, the previous proof becomes not entirely correct (probably it may be improved by regarding relative adjoint functors, but I don't think it's relevant to this question). However, there is a straightforward proof of this fact: $$(g\circ f=h\circ f)\Rightarrow(g\circ f\circ s_i=h\circ f\circ s_i)\Rightarrow(g\circ s_i\circ f_i=h\circ s_i\circ f_i)\Rightarrow(g\circ s_i=h\circ s_i)\Rightarrow(g\circ h),$$ where $$g$$ and $$h$$ are arbitrary morphisms of $$\mathbf{C}$$ with domain $$B$$ and $$s_i$$, $$i\in I$$, are canonical injections of coproducts.
As for the finite $$I$$ in your second question: there is no such counterexamples. An abelian category is always finitely complete, finitely cocomplete and finite coproducts coincide with finite products, hence if $$I$$ is finite, then the $$I$$-coproduct functor of an abelian (or even additive) category is both left- and right-adjoint, hence exact. So the axiom AB4 requires only that the infinite coproducts of monomorphisms should be monomorphisms.