-
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.
-
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.
-
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.
-
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.
-
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.
-
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.
-
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 $r\circ s=\text{id}_{B}$.