Composition in the Arrow category From a category $\mathcal{C}$ we can construct its arrow category $\text{Ar}(\mathcal{C})$, where objects are morphisms and arrows are commutative squares.
But what happens with arrow composition? This does not correspond directly to a monoidal category structure since not every two arrows can be composed to give rise to a third one.


*

*Where does the composition on arrows enters on the structure (besides the composition of morphisms between arrows themselves, that are pair of arrows)?

*Can the composition be expressed inside this category?


Thanks!

One possible way to "see" composition of arrows of the original category $\mathcal{C}$ in $\operatorname{Ar}(\mathcal{C})$ is the following:
Suppose $f:x\to y$ and $g:y\to z$, let $1_x,1_y,1_z\in\operatorname{Ar}(\mathcal{C})$ be the identity morphisms of $x,y,z$ respectively, then $f$ gives a morphism $1_f:1_x\to1_y$ in the obvious way, and similarly for $g$. Then
$$1_{g\circ f} = 1_g\circ1_f.$$
