Subsection 9.2.6 (01BY): Morphisms Corepresentably Fully Faithful on Cores

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

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is corepresentably fully faithful on cores if the following equivalent conditions are satisfied:

The $1$-morphism $f$ is corepresentably full on cores (Definition 9.2.5.1.1) and corepresentably faithful on cores (Definition 9.2.1.1.1).
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 fully faithful.

In detail, $f$ is corepresentably fully faithful on cores if the conditions in Remark 9.2.4.1.2 and Remark 9.2.5.1.2 hold:

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 $.
For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $2$-isomorphism
of $\mathcal{C}$, there exists a $2$-isomorphism
of $\mathcal{C}$ such that we have an equality
of pasting diagrams in $\mathcal{C}$, i.e. such that we have
\[ \beta =\alpha \mathbin {\star }\text{id}_{f}. \]

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