In detail, $f$ is corepresentably faithful on cores if, for all diagrams in $\mathcal{C}$ of the form
if $\alpha $ and $\beta $ are $2$-isomorphisms and we have
\[ \alpha \mathbin {\star }\text{id}_{f}=\beta \mathbin {\star }\text{id}_{f}, \]
then $\alpha =\beta $.