A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably fully faithful1 if the following equivalent conditions are satisfied:
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1.
The $1$-morphism $f$ is representably faithful (Definition 13.1.1.1.1) and representably full (Definition 13.1.2.1.1).
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2.
For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the functor
\[ f_{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (X,A\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (X,B\webright ) \]given by postcomposition by $f$ is fully faithful.
- 1Further Terminology: Also called simply a fully faithful morphism, based on Item 1 of Example 13.1.3.1.3.
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1.
For all diagrams in $\mathcal{C}$ of the form
if we have
\[ \operatorname {\mathrm{id}}_{f}\mathbin {\star }\alpha =\operatorname {\mathrm{id}}_{f}\mathbin {\star }\beta , \]then $\alpha =\beta $.
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2.
For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$ and each $2$-morphism
of $\mathcal{C}$, there exists a $2$-morphism
of $\mathcal{C}$ such that we have an equality
of pasting diagrams in $\mathcal{C}$, i.e. such that we have
\[ \beta =\operatorname {\mathrm{id}}_{f}\mathbin {\star }\alpha . \] -
1.
Representably Fully Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The representably fully faithful morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the fully faithful functors; see Chapter 11: Categories, Item 6 of Proposition 11.6.3.1.2.
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2.
Representably Fully Faithful Morphisms in $\boldsymbol {\mathsf{Rel}}$. The representably fully faithful morphisms of $\boldsymbol {\mathsf{Rel}}$ coincide (Chapter 8: Relations, Item 3 of Proposition 8.4.8.1.1) with the representably full morphisms in $\boldsymbol {\mathsf{Rel}}$, which are characterised in Chapter 8: Relations, Item 2 of Proposition 8.4.8.1.1.
13.1.3 Representably Fully Faithful Morphisms
Let $\mathcal{C}$ be a bicategory.
In detail, $f$ is representably fully faithful if the conditions in Remark 13.1.1.1.2 and Remark 13.1.2.1.2 hold:
Here are some examples of representably fully faithful morphisms.
