Let $\mathcal{C}$ be a category, and let $(\varphi, \psi)$ be an arrow in the arrow category $\mathcal{C}^\rightarrow$, where $\varphi : a \rightarrow a'$, $\psi : b \rightarrow b'$, and there exist $f : a \rightarrow b$, $g : a' \rightarrow b'$ such that the corresponding square commutes (that is, $\psi \circ f = g \circ \varphi$). Let's also assume $(\varphi, \psi)$ is monic in $\mathcal{C}^\rightarrow$. What does it tell us about $\varphi, \psi$ in $\mathcal{C}$?
First, I managed to prove the following: if $\xi_1, \xi_2 : c \rightarrow a, \chi_1, \chi_2 : d \rightarrow b$ are such that $(\xi_1, \chi_1)$ and $(\xi_2, \chi_2)$ are arrows in $\mathcal{C}^\rightarrow$ and such that $\varphi \circ \xi_1 = \varphi \circ \xi_2$ and $\psi \circ \chi_1 = \psi \circ \chi_2$, then $\xi_1 = \xi_2$ and $\chi_1 = \chi_2$:
That is, $\varphi$ and $\psi$ are monomorphisms if we restrict our arrows in $\mathcal{C}$ to the ones that make it into arrows in the arrow category.
Can we do better?
Let's take arbitrary $\xi_1, \xi_2 : c \rightarrow a$, also take $c$ as $d$ (and hence $h = \text{id}_c$), and take $\chi_1 = f \circ \xi_1, \chi_2 = f \circ \xi_2$. Then it follows that, in particular, $\varphi \circ \xi_1 = \varphi \circ \xi_2 \Rightarrow \xi_1 = \xi_2$, hence $\varphi$ is monic in $\mathcal{C}$.
Can we do a similar trick to prove $\psi$ is monic?
The best I could come up with is the following. Assume $\mathcal{C}$ has initial objects, take $c = 0$ and arbitrary $\chi_1, \chi_2 : d \rightarrow b$:
Then the leftmost square commutes, and by a similar argument we see $\psi$ is monic.
Does it make sense? If so, can I do the $\psi$ part without assuming initial objects?
Best Answer
Suppose that $C$ has an initial object $0$. Then, the codomain functor $C^{\rightarrow} \to C$ has a left adjoint sending each object $X$ of $C$ to the unique arrow $0 \to X$, and so it preserves monomorphisms.
Your proof that the codomain functor preserves monomorphisms is just a special case of the general proof of the fact that right adjoint functors preserve monomorphisms.
Also, regardless of the existence of an initial object in $C$, the domain functor $C^{\rightarrow} \to C$ always has a left adjoint sending each object of $C$ to its identity arrow. And again, your proof that the domain functor preserves monomorphisms is just a special case of the general proof for right adjoint functors.