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!

The arrow category is equipped with two functors $s, t : \text{Ar}(C) \to C$ giving the source and target of an arrow. Composition is a functor
$$\text{Ar}(C) \times_{\text{Ob}(C)} \text{Ar}(C) \to \text{Ar}(C)$$
where the LHS is a (2-)pullback, with one of the maps $\text{Ar}(C) \to \text{Ob}(C)$ being source and one being target. This expresses precisely that only morphisms with compatible sources and targets can be composed. 
