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