Arrow Category; Monomorphisms and Pushouts

Question

Let $\mathcal{C}^\rightarrow$ denote the arrow category of $\mathcal{C}$. I wonder about basic notions such as monomorphisms and pushouts for the general case of an arbitrary base category $\mathcal{C}$ but I also wonder whether one only gets nice results for certain wellbehaved categories such as maybe Sets, Graphs, ...?

Monomorphisms

It appears that two monos $f_1$ and $f_2$ from $\mathcal{C}$ form a mono $(f_1,f_2)$ in $\mathcal{C}^\rightarrow$. But that a mono $(f_1,f_2)$ in $\mathcal{C}^\rightarrow$ may not need to consist of two monos $f_1$ and $f_2$ from $\mathcal{C}$ (at least the proof obligation seems to indicate this). I wonder if for this reverse direction further obvious requirements are needed; e.g. that the arrow objects (say $k_1:A_1\rightarrow B_1$ and $k_2:A_2\rightarrow B_2$ such that $f_1:A_1\rightarrow B_1$ and $f_2:A_2\rightarrow B_2$) are monos (while this also seems to not help here).

Pushouts

The problem above about monos makes me wonder whether pushouts in $\mathcal{C}^\rightarrow$ can actually be constructed componentwise (or if they can be constructed at all (for all inputs)). That is, if $(f_1,f_2):(k_1:A_1\rightarrow B_1)\rightarrow(k_2:A_2\rightarrow B_2)$ and $(g_1,g_2):(k_1:A_1\rightarrow B_1)\rightarrow(k_3:A_3\rightarrow B_3)$ whether $(f'_1,f'_2):(k_2:A_2\rightarrow B_2)\rightarrow(k_4:A_4\rightarrow B_4)$ and $(g'_1,g'_2):(k_3:A_3\rightarrow B_3)\rightarrow(k_4:A_4\rightarrow B_4)$ can be constructed by constructing $f'_i$ and $g'_i$ for $f_i$ and $g_i$ ($i\in\{1,2\}$). While these four morphisms $f_i',g_i'$ can be constructed using this approach (when $\mathcal{C}$ has pushouts), the morphism $k_4$ needs to be derived then and whether this is always (uniquely) possible seems questionable to me.

Answer 1

As MPos mentioned in the comments, the answer is yes in most categories of interest. Specifically, as stated, we can recognize $\mathcal{C}^\to$ as the functor category $Fun([1], \mathcal{C})$, where $[1] = 0 \to 1$. In functor categories $Fun(A, \mathcal{C})$, if we know that $\mathcal{C}$ admits all (co)limits of shape $\mathcal{D}$, then we know that any functor $F :\mathcal{D} \to Fun(A, \mathcal{C})$ admits a (co)limit formed pointwise/objectwise. To see this, it will be useful to think of $F$ as a functor $A \times \mathcal{D} \to \mathcal{C}$ (this can be done by "currying" variables):

1. For each $a \in A$, calculate the (co)limit of the diagram $F(a, -) : \mathcal{D} \to \mathcal{C}$,
2. Check that the universal property of (co)limits induces maps $\mathsf{(co)lim}F(a, -) \to \mathsf{(co)lim}F(a', -)$ for each $a \to a' \in A$; this defines a functor $A \to \mathcal{C}$,
3. Check that this functor is a (co)limit in $Fun(A, \mathcal{C})$.

Thus, any (co)limit of shape $\mathcal{D}$ in $Fun(A, \mathcal{C})$ must be pointwise. In particular, if $\mathcal{C}$ admits all pushouts, this answers your second question. Now, a morphism $f : X \to Y$ is a monomorphism if and only if the following diagram is a pullback (this is easy to directly verify)

Thus, provided $\mathcal{C}$ admits all pullbacks, a morphism $(f_1,f_2)$ is monic in $\mathcal{C}^\to$ iff $f_1$ and $f_2$ are monic in $\mathcal{C}$. Many categories of interest admit all pushouts and pullbacks (e.g. $\mathsf{Grp},\mathsf{Top},\mathsf{CRing},\mathsf{Ab},\mathsf{Set}$, ...), and the ones that don't (say, the category of manifolds) you're less likely to be working with categorically anyways.

For an example of a category $\mathcal{C}$ and morphism in $\mathcal{C}^\to$ which is monic but not pointwise monic, see https://math.stackexchange.com/a/95196. As you can see, it's a bit of a contrived example.