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 $.


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