9.1.5 Morphisms Representably Full on Cores

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

Definition 9.1.5.1.1 ▶ Morphisms Representably Full on Cores

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably full 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 full.

Remark 9.1.5.1.2 ▶ Unwinding Definition 9.1.5.1.1

In detail, $f$ is representably full on cores if, 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

\[ \beta =\text{id}_{f}\mathbin {\star }\alpha . \]

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