9.2.4 Morphisms Corepresentably Faithful on Cores

Let $\mathcal{C}$ be a bicategory.

Definition 9.2.4.1.1 ▶ Morphisms Corepresentably Faithful on Cores

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is corepresentably faithful on cores if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor

\[ f^{*}\colon \mathsf{Core}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (B,X\webright )\webright )\to \mathsf{Core}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (A,X\webright )\webright ) \]

given by precomposition by $f$ is faithful.

Remark 9.2.4.1.2 ▶ Unwinding Definition 9.2.4.1.1

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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