The Clowder Project

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

1. Interaction With Postcomposition. The following conditions are equivalent:
   1. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is faithful.
   2. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
      \[ F_{*} \colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright ) \to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]

      is faithful.

   3. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a representably faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 9: Types of Morphisms in Bicategories, Definition 9.1.1.1.1.
2. Interaction With Precomposition I. Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
   1. If $F$ is faithful, then the precomposition functor
      \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

      can fail to be faithful.

   2. Conversely, if the precomposition functor
      \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

      is faithful, then $F$ can fail to be faithful.

3. Interaction With Precomposition II. If $F$ is essentially surjective, then the precomposition functor
   \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

   is faithful.

4. Interaction With Precomposition III. The following conditions are equivalent:
   1. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
      \[ F^{*} \colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright ) \to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

      is faithful.

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

      is conservative.

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

      is monadic.

   4. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a corepresentably faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 9: Types of Morphisms in Bicategories, Definition 9.2.1.1.1.
   5. The components
      \[ \eta _{G}\colon G\Longrightarrow \text{Ran}_{F}\webleft (G\circ F\webright ) \]

      of the unit

      \[ \eta \colon \text{id}_{\mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )}\Longrightarrow \text{Ran}_{F}\circ F^{*} \]

      of the adjunction $F^{*}\dashv \text{Ran}_{F}$ are all monomorphisms.

   6. The components
      \[ \epsilon _{G}\colon \text{Lan}_{F}\webleft (G\circ F\webright )\Longrightarrow G \]

      of the counit

      \[ \epsilon \colon \text{Lan}_{F}\circ F^{*}\Longrightarrow \text{id}_{\mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )} \]

      of the adjunction $\text{Lan}_{F}\dashv F^{*}$ are all epimorphisms.

   7. The functor $F$ is dominant (Definition 8.6.1.1.1), i.e. every object of $\mathcal{D}$ is a retract of some object in $\mathrm{Im}\webleft (F\webright )$:
      • For each $B\in \text{Obj}\webleft (\mathcal{D}\webright )$, there exist:
        • An object $A$ of $\mathcal{C}$;
        • A morphism $s\colon B\to F\webleft (A\webright )$ of $\mathcal{D}$;
        • A morphism $r\colon F\webleft (A\webright )\to B$ of $\mathcal{D}$;
        such that $r\circ s=\text{id}_{B}$.