nLab cofinally small category

Definition

A category $C$ is cofinally small if there is a small category $C_0$ and a cofinal functor $C_0 \to C$.

Of course, if we choose the terminology of final functors, this property is called finally small.

If $C$ is locally small, then $C$ is cofinally small if and only if it has a small cofinal full subcategory (i.e. the inclusion functor is cofinal).