In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably full on cores if, for each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and each natural isomorphism
there exists a natural isomorphism such that we have an equality of pasting diagrams in $\mathsf{Cats}_{\mathsf{2}}$, i.e. such that we have
\[ \beta =\text{id}_{F}\mathbin {\star }\alpha . \]