A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably fully faithful on cores if the following equivalent conditions are satisfied:
-
1.
The $1$-morphism $f$ is representably faithful on cores (Definition 13.1.5.1.1) and representably full on cores (Definition 13.1.4.1.1).
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2.
For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, 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 fully faithful.
-
1.
For all diagrams in $\mathcal{C}$ of the form
if $\alpha $ and $\beta $ are $2$-isomorphisms and we have
\[ \operatorname {\mathrm{id}}_{f}\mathbin {\star }\alpha =\operatorname {\mathrm{id}}_{f}\mathbin {\star }\beta , \]then $\alpha =\beta $.
-
2.
For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$ 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
\[ \beta =\operatorname {\mathrm{id}}_{f}\mathbin {\star }\alpha . \]
13.1.6 Morphisms Representably Fully Faithful on Cores
Let $\mathcal{C}$ be a bicategory.
In detail, $f$ is representably fully faithful on cores if the conditions in Remark 13.1.4.1.2 and Remark 13.1.5.1.2 hold:
