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 full.
   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 full.

   3. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a representably full morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 9: Types of Morphisms in Bicategories, Definition 9.1.2.1.1.
2. Interaction With Precomposition I. If $F$ is full, 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 full.

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

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

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

   is full (and also faithful by Item 3 of Proposition 8.5.1.1.2).

5. Interaction With Precomposition IV. 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 full.

   2. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a corepresentably full morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 9: Types of Morphisms in Bicategories, Definition 9.2.1.1.1.
   3. 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 retractions/split epimorphisms.

   4. 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 sections/split monomorphisms.

   5. For each $B\in \text{Obj}\webleft (\mathcal{D}\webright )$, there exist:
      • An object $A_{B}$ of $\mathcal{C}$;
      • A morphism $s_{B}\colon B\to F\webleft (A_{B}\webright )$ of $\mathcal{D}$;
      • A morphism $r_{B}\colon F\webleft (A_{B}\webright )\to B$ of $\mathcal{D}$;
      satisfying the following condition:
      • For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each pair of morphisms
        \begin{align*} r & \colon F\webleft (A\webright ) \to B,\\ s & \colon B \to F\webleft (A\webright ) \end{align*}

        of $\mathcal{D}$, we have

        \[ \webleft [\webleft (A_{B},s_{B},r_{B}\webright )\webright ]=\webleft [\webleft (A,s,r\circ s_{B}\circ r_{B}\webright )\webright ] \]

        in $\int ^{A\in \mathcal{C}}h^{B'}_{F_{A}}\times h^{F_{A}}_{B}$.