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.

Item 1: Characterisations
Omitted.


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