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 pseudoepic.
   2. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the functor
      \[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

      given by precomposition by $F$ is pseudomonic.

   3. We have an isococomma square of the form
      
      in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
2. Dominance. If $F$ is pseudoepic, then $F$ is dominant (Definition 8.6.1.1.1).