The Clowder Project

Let $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ be functors.

1. Characterisations. The following conditions are equivalent:
   1. The functor $F$ is fully faithful.
   2. We have a pullback square
      
      in $\mathsf{Cats}$.
2. Interaction With Composition. If $F$ and $G$ are fully faithful, then so is $G\circ F$.
3. Conservativity. If $F$ is fully faithful, then $F$ is conservative.
4. Essential Injectivity. If $F$ is fully faithful, then $F$ is essentially injective.
5. Interaction With Co/Limits. If $F$ is fully faithful, then $F$ reflects co/limits.
6. Interaction With Postcomposition. The following conditions are equivalent:
   1. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is fully 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 fully faithful.

   3. The functor $F\colon \mathcal{C}\to \mathcal{D}$ is a representably fully faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 11: Types of Morphisms in Bicategories, Definition 11.1.3.1.1.
7. Interaction With Precomposition I. If $F$ is fully 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 fully faithful.

8. 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 fully faithful, then $F$ can fail to be fully faithful (and in fact it can also fail to be either full or faithful).

9. 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 fully faithful.

10. 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 fully faithful.

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

       is fully faithful.

    3. The functor
       \[ \text{Lan}_{F}\colon \mathsf{Fun}\webleft (\mathcal{C},\mathsf{Sets}\webright ) \to \mathsf{Fun}\webleft (\mathcal{D},\mathsf{Sets}\webright ) \]

       is fully faithful.

    4. The functor $F$ is a corepresentably fully faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 11: Types of Morphisms in Bicategories, Definition 11.2.3.1.1.
    5. The functor $F$ is absolutely dense.
    6. 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 isomorphisms.

    7. 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 isomorphisms.

    8. The natural transformation
       \[ \alpha \colon \text{Lan}_{h_{F}}\webleft (h^{F}\webright )\Longrightarrow h \]

       with components

       \[ \alpha _{B',B}\colon \int ^{A\in \mathcal{C}}h^{B'}_{F_{A}}\times h^{F_{A}}_{B}\to h^{B'}_{B} \]

       given by

       \[ \alpha _{B',B}\webleft (\webleft [\webleft (\phi ,\psi \webright )\webright ]\webright )=\psi \circ \phi \]

       is a natural isomorphism.

    9. 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 conditions:
       1. The triple $\webleft (F\webleft (A_{B}\webright ),r_{B},s_{B}\webright )$ is a retract of $B$, i.e. we have $r_{B}\circ s_{B}=\text{id}_{B}$.
       2. For each morphism $f\colon B'\to B$ of $\mathcal{D}$, we have
          \[ \webleft [\webleft (A_{B},s_{B'},f\circ r_{B'}\webright )\webright ]=\webleft [\webleft (A_{B},s_{B}\circ f,r_{B}\webright )\webright ] \]

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