Arrow category ($[2,\mathcal{C}]$)

The arrow category of $\mathcal{C}$ has morphisms $A_0\to A_1$ of $\mathcal{C}$ as objects and commutative squares as morphisms. It is the functor category $[2,\mathcal{C}]$, where $2$ is the category with one nonidentity arrow.

The category of injective functions is the full subcategory of $[2,\mathbf{Set}]$ whose objects are injective functions. It is cartesian closed: products are computed pointwise, and exponentiating an injection by any arrow again gives an injection.