The Clowder Project

Here are some examples of strict monomorphisms.

1. Strict Monomorphisms in $\mathsf{Cats}_{\mathsf{2}}$. The strict monomorphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the functors which are injective on objects and injective on morphisms; see Chapter 9: Preorders, Item 1 of Proposition 9.7.2.1.2.
2. Strict Monomorphisms in $\textbf{Rel}$. The strict monomorphisms in $\textbf{Rel}$ are characterised in Chapter 6: Relations, Proposition 6.3.7.1.1.