Comma category

Summary

Let $S \colon \mathcal{A} \to \mathcal{C}$ and $T \colon \mathcal{B} \to \mathcal{C}$ be functors. The comma category $(S \downarrow T)$ has as objects the morphisms $S(A) \to T(B)$ of $\mathcal{C}$. Its morphisms are the pairs of morphisms that make the evident square commute. It turns morphisms of one category into objects of a new category. Slice categories, arrow categories and categories of elements are special comma categories. Universal properties and adjoint functors can be expressed as initial or terminal objects of comma categories.