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

\[ \beta =\text{id}_{F}\mathbin {\star }\alpha . \]

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