The Clowder Project

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

1. Characterisations. The following conditions are equivalent:
   1. The functor $F$ is a monomorphism of categories.
   2. The functor $F$ is injective on objects and morphisms, i.e. $F$ is injective on objects and the map
      \[ F\colon \textup{Mor}\webleft (\mathcal{C}\webright )\to \textup{Mor}\webleft (\mathcal{D}\webright ) \]

      is injective.