Category of elements

Remarks

Composition is well defined because $F(g \circ f)(x) = F(g)(F(f)(x))$. It is the comma category $(\ast \downarrow F)$, where $\ast$ is a one-point set, and the map $(A, x) \mapsto A$ is a projection functor $\int F \to \mathcal{C}$. Example: let $G$ be a group seen as a category with one object, and let $F$ be a set $X$ with a group action of $G$. Then $\int F$ has the elements of $X$ as objects and a morphism $g \colon x \to y$ for each $g \in G$ with $g \cdot x = y$; it is the action groupoid. Non-example: if $g \cdot x \ne y$, then $g$ is not a morphism from $x$ to $y$ in $\int F$, although it is a morphism of $G$. For a contravariant functor $P \colon \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$, the same construction is usually written with morphisms $f \colon A \to B$ such that $P(f)(y) = x$.