11.1.4 Morphisms Representably Faithful on Cores

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

Definition 11.1.4.1.1 ▶ Morphisms Representably Faithful on Cores

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

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

given by postcomposition by $f$ is faithful.

Remark 11.1.4.1.2 ▶ Unwinding Definition 11.1.4.1.1

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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