In detail, $f$ is representably fully faithful if the conditions in Remark 11.1.1.1.2 and Remark 11.1.2.1.2 hold:
-
For all diagrams in $\mathcal{C}$ of the form
if we have
\[ \text{id}_{f}\mathbin {\star }\alpha =\text{id}_{f}\mathbin {\star }\beta , \]then $\alpha =\beta $.
-
For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $2$-morphism of $\mathcal{C}$, there exists a $2$-morphism
of $\mathcal{C}$ such that we have an equality
of pasting diagrams in $\mathcal{C}$, i.e. such that we have
\[ \beta =\text{id}_{f}\mathbin {\star }\alpha . \]
