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 pseudomonic.
   2. The functor $F$ satisfies the following conditions:
      1. The functor $F$ is faithful, i.e. for each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the action on morphisms
         \[ F_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \]

         of $F$ at $\webleft (A,B\webright )$ is injective.

      2. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the restriction
         \[ F^{\textup{iso}}_{A,B} \colon \textup{Iso}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Iso}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \]

         of the action on morphisms of $F$ at $\webleft (A,B\webright )$ to isomorphisms is surjective.

   3. We have an isocomma square of the form
      
      in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
   4. We have an isocomma square of the form
      
      in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
   5. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition¹ functor
      \[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]

      is pseudomonic.

2. Conservativity. If $F$ is pseudomonic, then $F$ is conservative.
3. Essential Injectivity. If $F$ is pseudomonic, then $F$ is essentially injective.