The Clowder Project

Here are some examples of representably fully faithful morphisms.

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