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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 fully faithful.
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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.
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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.
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The functor $F$ is a corepresentably fully faithful morphism in $\mathsf{Cats}_{\mathsf{2}}$ in the sense of Chapter 11: Graphs, Definition 11.2.3.1.1.
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The functor $F$ is absolutely dense.
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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 isomorphisms.
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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 isomorphisms.
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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.
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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 conditions:
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The triple $\webleft (F\webleft (A_{B}\webright ),r_{B},s_{B}\webright )$ is a retract of $B$, i.e. we have $r_{B}\circ s_{B}=\text{id}_{B}$.
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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 $\int ^{A\in \mathcal{C}}h^{B'}_{F_{A}}\times h^{F_{A}}_{B}$.