11.2.6 Morphisms Corepresentably Fully Faithful on Cores

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

Definition 11.2.6.1.1 ▶ Morphisms Corepresentably Fully Faithful on Cores

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

  1. The $1$-morphism $f$ is corepresentably full on cores (Definition 11.2.5.1.1) and corepresentably faithful on cores (Definition 11.2.1.1.1).
  2. 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.

Remark 11.2.6.1.2 ▶ Unwinding Definition 11.2.6.1.1

In detail, $f$ is corepresentably fully faithful on cores if the conditions in Remark 11.2.4.1.2 and Remark 11.2.5.1.2 hold:

  1. 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 $.

  2. 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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