Category of elements

Summary

The category of elements of a functor $F \colon \mathcal{C} \to \mathbf{Set}$ has as objects the pairs $(A, x)$ with $x \in F(A)$. Its morphisms are the morphisms of $\mathcal{C}$ that carry one chosen element to the other. It turns a set-valued functor into a category lying over $\mathcal{C}$, and it is a special comma category. It is used to write every presheaf as a colimit of representable functors, and it is the simplest case of the Grothendieck construction.