In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably fully faithful on cores if it satisfies the conditions in Remark 8.7.7.1.2 and Remark 8.7.8.1.2, i.e.:
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For all diagrams of the form
if $\alpha $ and $\beta $ are natural isomorphisms and we have
\[ \alpha \mathbin {\star }\text{id}_{F}=\beta \mathbin {\star }\text{id}_{F}, \]then $\alpha =\beta $.
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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
\[ \beta =\alpha \mathbin {\star }\text{id}_{F}. \]
