The Clowder Project

Here are some examples of representably fully faithful morphisms.

1. 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 9: Categories, Item 6 of Proposition 9.6.3.1.2.
2. 2. Representably Fully Faithful Morphisms in $\mathbf{\text{Rel}}$. The representably fully faithful morphisms of $\mathbf{\text{Rel}}$ coincide (Chapter 6: Relations, Item 3 of Proposition 6.3.8.1.1) with the representably full morphisms in $\mathbf{\text{Rel}}$, which are characterised in Chapter 6: Relations, Item 2 of Proposition 6.3.8.1.1.