Let $\mathcal{A}$ be (an abelian, I don't think it should matter?) category. Then let $\operatorname{Mor}(\mathcal{A})$ be the category of morphisms, that is the category with objects given by morphisms of $\mathcal{A}$ and morphisms $f \rightarrow g$ given by commuting squares of $\mathcal{A}$.
So now let $f: A \rightarrow B,\ g: C \rightarrow D$ be two objects of the arrow category. Is it true that an epimorphism (mono-) $\phi: f \rightarrow g$ is a pair of morphisms $\phi_1: A \rightarrow C$, and $\phi_2: B\rightarrow D$ such that the resultign square commutes and the $\phi_i$ are epimorphisms (mono-) in $\mathcal{A}$?
If not, how can epi/monomorphisms of the arrow category be characterized?
Best Answer
I’ll deal with the case of monomorphisms. A dual argument works for epimorphisms.
For an abelian category, what you ask about is true, since $\phi$ has a kernel $\ker(\phi_1)\to\ker(\phi_2)$, which is zero iff $\phi_1$ and $\phi_2$ are monomorphisms.
For a general category it’s not true. Take the simplest category with a non-monomorphism: there are precisely three objects $X$, $Y$ and $Z$ with two maps $\alpha,\beta:X\to Y$ and one map $\gamma:Y\to Z$ such that $\gamma\alpha=\gamma\beta$ and no other maps except identity maps. So $\gamma$ is not a monomorphism.
However consider the map of morphisms $\phi$ from $\alpha$ to $\gamma$ with $\phi_1=\alpha$ and $\phi_2=\gamma$.
This is a monomorphism, since no object of the morphism category has more than one map to $\alpha$.