In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably faithful on cores if, given a diagram of the form
if $\alpha $ and $\beta $ are natural isomorphisms and we have
\[ \alpha \mathbin {\star }\text{id}_{F}=\beta \mathbin {\star }\text{id}_{F}, \]
then $\alpha =\beta $.