The question is regarding the following axioms
AB$4$. If $(f_i\colon A_i\to B_i)_{i \in I}$ is a family of monomorphisms in an abelian category, then the induced $\bigoplus_{i \in I} f_i\colon \bigoplus_{i \in I} A_i\to \bigoplus_{i \in I} B_i$ is also a monomorphism.
AB$5$. In an abelian category, for a directed family $(A_i)_{i \in I}$ of subobjects of $A$ and a subobject $B$ of $A$, we have $(\sum_{i \in I} A_i)\cap B = \sum_{i \in I} (A_i\cap B)$.
I'm trying to follow the following proof from N.Popescu's book Abelian categories with applications to rings and modules:
However, I get stuck at the end, not seeing why $v_F$ is a monomorphism. The way I understand it, if $(l_F)_i\colon X_i \to \coprod_{i \in F} X_i$ and $l_i\colon X_i\to \coprod_{i \in I} X_i$ are canonical injections, then $v_F$ is the unique morphism such that $v_F\circ (l_F)_i = l_i$ for all $i \in F$. Why would it be monic?
I understand, however, why $f_F$ is monic. It is because direct sums of finite families of monomorphisms coincides their products, hence they are monomorphisms as limits commute with limits. But this logic doesn't apply to $v_F$ since it has possibly infinite set $I$ and the colimit of the constant discrete category is not the same as the limit of the constant connected category, so we can't use the trick of reducing the universal property morphism to the morphism induced by colimits, as in connected case.
Special thanks to Captain Lama in the comment section!
Let $(l_{I\setminus F})_i\colon X_i\to \coprod_{i \in I\setminus F} X_i$. Form a coproduct $(\coprod_{i \in F} X_i)\sqcup (\coprod_{i \in F\setminus I} X_i)$ together with canonical injections $q_1\colon \coprod_{i \in F} X_i \to (\coprod_{i \in F} X_i)\sqcup (\coprod_{i \in F\setminus I} X_i)$ and $q_2 \colon \coprod_{i \in F} X_i \to (\coprod_{i \in F} X_i)\sqcup (\coprod_{i \in F\setminus I} X_i)$. Then, by the associativity theorem for coproducts, $(\coprod_{i \in F} X_i)\sqcup (\coprod_{i \in F\setminus I} X_i)$ is a coproduct of the family $(X_i)_{i \in I}$ together with the canonical injections $q_i\colon X_i \to (\coprod_{i \in F} X_i)\sqcup (\coprod_{i \in F\setminus I} X_i)$ where $q_i = q_1\circ (l_F)_i$ if $i \in F$ and $q_i = q_2\circ (l_{F\setminus I})_i$ otherwise. By uniqueness of coproduct, there is a unique isomorphism $u$ such that $l_i = u\circ q_i = u\circ q_1\circ (l_F)_i$ for all $i \in F$ and $l_i = u\circ q_2 \circ (l_{F\setminus I})_i$ for all $i \in F\setminus I$. In particular, $v_F\circ (l_F)_i = l_i = u\circ q_1 \circ (l_F)_i$ for all $i \in F$, so $v_F = u\circ q_1$ is a monomorphism as coproduct injections are monomorphisms in abelian categories.