In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably fully faithful on cores if it satisfies the conditions in Remark 9.8.4.1.2 and Remark 9.8.5.1.2, i.e.:
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For all diagrams of the form
with $\alpha $ and $\beta $ natural isomorphisms, if we have $\text{id}_{F}\mathbin {\star }\alpha =\text{id}_{F}\mathbin {\star }\beta $, then $\alpha =\beta $.
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For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and each natural isomorphism of $\mathcal{C}$, there exists a natural isomorphism
of $\mathcal{C}$ such that we have an equality
of pasting diagrams in $\mathsf{Cats}_{\mathsf{2}}$, i.e. such that we have
\[ \beta =\text{id}_{F}\mathbin {\star }\alpha . \]
