In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably full on cores if, for each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and each natural isomorphism
\[ \beta =\alpha \mathbin {\star }\text{id}_{F}. \]