nLab
representable fibered category

Given an object $X$ in a category $B$ the domain functor $(Y\stackrel{f}{\to }X)\mapsto Y$ from the slice category $B/X$ to $B$ is a fibered category (i.e. Grothendieck fibration).

Any fibered category isomorphic to the $\mathrm{dom}:B/X\to B$ is said to be representable. This is because under the Grothendieck construction representable fibered categories correspond precisely to representable functors $B^{\mathrm{op}}\to \mathrm{Set}\hookrightarrow \mathrm{Cat}$: the category $B/X$ is the category of elements of the representable functor $B(-,X)$.