The Clowder Project

Let $f\colon A\to B$ be a $1$-morphism of $\mathcal{C}$.

1. Characterisations. The following conditions are equivalent:
   1. The morphism $f$ is pseudomonic.
   2. The morphism $f$ is representably full on cores and representably faithful.
   3. We have an isocomma square of the form
      
      in $\mathcal{C}$ up to equivalence.
2. Interaction With Cotensors. If $\mathcal{C}$ has cotensors with $\mathbb {1}$, then the following conditions are equivalent:
   1. The morphism $f$ is pseudomonic.
   2. We have an isocomma square of the form
      
      in $\mathcal{C}$ up to equivalence.