Very early on in getting to grips with category theory, we meet various ways of getting new categories from a given category $\mathscr{C}$ -- take the dual, quotient by a congruence on arrows, slice over or under an object, form the arrow category $\mathscr{C}^{\to}$, ...
Now, some of those definitions are immediately accompanied by examples showing that we in fact are already familiar with categories of these new kinds. Thus, for just one example, the coslice category $\mathbf{Set}/1$, for $1$ terminal, is (tantamount to) the category $\mathbf{Set_*}$ of pointed sets.
But, unless I've just not been concentrating, the introduction of the idea of an arrow category $\mathscr{C}^{\to}$ is not usually accompanied by a nice example along similar lines, "And look, the arrow category $\mathbf{SomeCat}^{\to}$ is (tantamount) to $\mathbf{FamiliarCat}$ ..."
Sure, later we find that the arrow category $\mathscr{C}^{\to}$ is a certain comma category, relates to a category of functors $2 \to \mathscr{C}$ and other things. But these general excitements are not the kind of thing I'm after! Ideally, I'm looking for nice particular examples of categories that -- even a few lectures in -- look to be naturally arising categories that turn out to be arrow categories in thin disguise. Any offers?
I want to say that this is not really the point of taking arrow categories. By analogy, in group theory when you learned about the commutator subgroup $[G, G]$, the point was not that this was a construction that let you reproduce familiar groups, it was that the commutator subgroup of a group is useful for understanding how it behaves. Similarly here.
Here is a half-example. If $A$ is an abelian category, then the arrow category $\text{Arr}(A)$ is the category of two-term chain complexes in $A$. There are two naturally defined functors $\text{Arr}(A) \to A$ given by taking the kernel or the cokernel of a morphism, and it's an interesting question to ask what their derived functors are. To my mind the easiest way to remember the answer to this question is to remember the interpretation in terms of two-term chain complexes.