Subsection 9.1.10 (01AU): Pseudomonic Morphisms

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Let $\mathcal{C}$ be a bicategory.

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is pseudomonic if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor

\[ f_{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (X,A\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (X,B\webright ) \]

given by postcomposition by $f$ is pseudomonic.

In detail, a $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is pseudomonic if it satisfies the following conditions:

For all diagrams in $\mathcal{C}$ of the form

if we have

\[ \text{id}_{f}\mathbin {\star }\alpha =\text{id}_{f}\mathbin {\star }\beta , \]

then $\alpha =\beta $.
For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $2$-isomorphism
of $\mathcal{C}$, there exists a $2$-isomorphism
of $\mathcal{C}$ such that we have an equality
of pasting diagrams in $\mathcal{C}$, i.e. such that we have
\[ \beta =\text{id}_{f}\mathbin {\star }\alpha . \]

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

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.
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.