I've been working with coproducts, and, as expected, dealing with a morphism to a coproduct is quite difficult. One thing that would help me is to work with morphisms that factor through their associated inclusions, so that is how I ended up with the following problem, which in certain well behaved categories seems to hold, but it is at the same time not obvious that it holds in other categories.
Say $A$ and $B$ are objects in an arbitrary category. Let $A \coprod B$ be their coproduct, with the respective inclusion morphisms $u_A$ and $u_B$.
Let $A'$ and $B'$ also be objects, $(A' \coprod B', u_{A'}, u_{B'})$ be their coproduct, and $f_A : A' \rightarrow A$ and $f_B : B' \rightarrow B$ be monomorphisms. Let $\varphi = (u_A \circ f_A) \coprod (u_B \circ f_B)$ be the unique morphism from $A' \coprod B'$ to $A \coprod B$ such that $\varphi \circ u_{A'} = u_A \circ f_A$ and $\varphi \circ u_{B'} = u_B \circ f_B$.
Finally, let $C$ be an object and $g : C \rightarrow A' \coprod B'$ be a morphism.
My question is: if $\varphi \circ g = u_A \circ g_A$ for some morphism $g_A : C \rightarrow A$, then should $g$ factor through $u_{A'}$? That is, should $g$ be equal to $u_{A'} \circ g_{A'}$ for some $g_{A'} : C \rightarrow A'$?
I've noticed that in the categories of sets, topological spaces, algebras, rings and vector spaces this property seems to hold, but I can't figure out if the same happens to other categories. Finding sufficient conditions on the category for this to happen would be pretty helpful too. Thanks!