If $c_0 \to c_1 \leftarrow c_0$ is a cocategory object in $C$, then by homming out of $c_\bullet$, one obtains a limit-preserving functor $C \to Cat$. Under reasonable conditions, the adjoint functor theorem conversely implies that all limit-preserving functors $C \to Cat$ are obtained in this way.

Examples

Many interval objects are cocategory objects. For example, the arrow category is a cocategory object in $Cat$.

In $Set$, every cocategory object is a coproduct of copies of the trivial cocategory object $1 \to 1 \leftarrow 1$ and the cocategory object $1 \to 2 \leftarrow 1$ where the two points are distinct. These represent the discrete category functor and the codiscrete category functors $Set \to Cat$, respectively.