# Kerodon

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Lemma 8.4.5.10. Let $\kappa$ be an uncountable regular cardinal and let $\lambda$ be an uncountable cardinal of exponential cofinality $\geq \kappa$. Let $\operatorname{\mathcal{C}}$ be an essentially $\kappa$-small $\infty$-category, let $\operatorname{\mathcal{D}}$ be a locally $\lambda$-small $\infty$-category, and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which satisfies the following conditions:

$(0)$

The $\infty$-category $\operatorname{\mathcal{D}}$ admits $\kappa$-small colimits.

$(1)$

The functor $f$ is fully faithful.

$(2)$

Let $C$ be an object of $\operatorname{\mathcal{C}}$. Then the image $f(C) \in \operatorname{\mathcal{D}}$ corepresents a functor $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}^{<\lambda }$ which preserves $\kappa$-small colimits.

Then $f$ is isomorphic to the composition $F \circ h_{\bullet }$, for some fully faithful functor

$F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$

which preserves $\kappa$-small colimits. Moreover, the functor $F$ is uniquely determined up to isomorphism.

Proof. It follows from Variant 8.4.3.6 that $f$ is isomorphic to a composition $F \circ h_{\bullet }$, for some functor $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{D}}$ which preserves $\kappa$-small colimits (and that $F$ is uniquely determined up to isomorphism). To complete the proof, it will suffice to show that the functor $F$ is fully faithful. Since $\lambda$ has exponential cofinality $\geq \kappa$, the functor $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is locally $\lambda$-small (Corollary 5.4.8.9). For every pair of functors $\mathscr {G}, \mathscr {G}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, the functor $F$ induces a morphism of Kan complexes

$\theta _{\mathscr {G}, \mathscr {G}'}: \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) }( \mathscr {G}, \mathscr {G}' ) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F( \mathscr {G}), F( \mathscr {G}' ) ).$

By virtue of Corollary 8.3.5.8 (and Remark 8.3.5.9), we can promote the construction $(\mathscr {G}, \mathscr {G}') \mapsto \theta _{ \mathscr {G}, \mathscr {G}' }$ to a functor of $\infty$-categories

$\theta : \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \times \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}^{< \lambda } ).$

We wish to show that, for every pair of functors $\mathscr {G}, \mathscr {G}': \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, the morphism $\theta _{\mathscr {G}, \mathscr {G}'}$ is a homotopy equivalence. Let us first regard the functor $\mathscr {G}'$ as fixed. Since $F$ preserves $\kappa$-small colimits, it follows from Remark 7.4.5.15 that the functor

$\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa })^{\operatorname{op}} \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}^{< \lambda } ) \quad \quad \mathscr {G} \mapsto \theta _{ \mathscr {G}, \mathscr {G}'}$

preserves $\kappa$-small limits. By virtue of Corollary 8.4.3.12, it will suffice to show that $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence in the special case where $\mathscr {G} = h_{C}$ for some object $C \in \operatorname{\mathcal{C}}$. Let us now regard $\mathscr {G} = h_ C$ as fixed. Combining Example 8.4.5.2 with assumption $(2)$, we deduce that the functor

$\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}^{< \lambda } ) \quad \quad \mathscr {G}' \mapsto \theta _{ \mathscr {G}, \mathscr {G}'}$

preserves $\kappa$-small colimits. Invoking Corollary 8.4.3.12 again, we are reduced to proving that $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence in the special case where $\mathscr {G}' = h_{C'}$ for some object $C' \in \operatorname{\mathcal{C}}$. In this case, we have a commutative diagram of Kan complexes

$\xymatrix@R =50pt@C=50pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,C') \ar [dl] \ar [dr] & \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )}( \mathscr {G}, \mathscr {G}' ) \ar [rr]^{ \theta _{\mathscr {G}, \mathscr {G}'} } & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(\mathscr {G}), F( \mathscr {G}') ), }$

where the left vertical map is a homotopy equivalence by virtue of Yoneda's lemma (Theorem 8.3.3.13). It will therefore suffice to show that the right vertical map is a homtoopy equivalence. Since $F \circ h_{\bullet }$ is isomorphic to $f$, this is equivalent to the assertion that the natural map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,C') \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( f(C), f(C') )$ is a homotopy equivalence, which follows from assumption $(1)$. $\square$