The category of elements of a covariant functor or a presheaf

Let $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ be a functor. Its category of elements $\int F$ has:

• objects $(c,x)$ with $c\in \mathcal{C}$ and $x\in F(c)$;
• a morphism $(c,x)\to (d,y)$ given by a morphism $f:c\to d$ in $\mathcal{C}$ satisfying $F(f)(x)=y$.

The identity of $(c,x)$ is $1_c$, and composition is composition in $\mathcal{C}$. The functor laws of Covariant functor, identity functor, composite functor, and contravariant functor make the identity and composite equations hold, so these data satisfy Category, object, morphism, domain, codomain, identity, composition, and hom-collection.

For a presheaf $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$, its category of elements $\int P$ again has objects $(c,x)$ with $x\in P(c)$, while a morphism $(c,x)\to (d,y)$ is a morphism $f:c\to d$ in $\mathcal{C}$ satisfying

$x=P(f)(y)\mathrm{.}$

This reversed equation comes from the opposite category Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$; contravariant functoriality again supplies identities and composition. A universal element in the sense of Universal elements of covariant functors and presheaves is itself an object of the appropriate category of elements.