Item 1 (015P) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

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.

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