In detail, $f$ is corepresentably fully faithful if the conditions in Remark 9.2.1.1.2 and Remark 9.2.2.1.2 hold:
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For all diagrams in $\mathcal{C}$ of the form
if we have
\[ \alpha \mathbin {\star }\text{id}_{f}=\beta \mathbin {\star }\text{id}_{f}, \]then $\alpha =\beta $.
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For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $2$-morphism of $\mathcal{C}$, there exists a $2$-morphismof $\mathcal{C}$ such that we have an equalityof pasting diagrams in $\mathcal{C}$, i.e. such that we have\[ \beta =\alpha \mathbin {\star }\text{id}_{f}. \]