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