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 $.


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