Proof.
By virtue of Proposition 8.4.5.7, we may assume without loss of generality that $\widehat{\operatorname{\mathcal{C}}}$ is the smallest replete full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ which contains all representable functors and is closed under the formation of $K$-indexed colimits for $K \in \mathbb {K}$ (see Construction 8.4.5.5). For every pair of objects $\mathscr {G}, \mathscr {G}' \in \widehat{\operatorname{\mathcal{C}}}$, the functor $F$ induces a morphism of Kan complexes
\[ \theta _{\mathscr {G}, \mathscr {G}'}: \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \mathscr {G}, \mathscr {G}' ) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F( \mathscr {G}), F( \mathscr {G}' ) ). \]
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
\[ \theta : \widehat{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}^{< \kappa } ). \]
We wish to show that, for every pair of objects $\mathscr {G}, \mathscr {G}' \in \widehat{\operatorname{\mathcal{C}}}$, the morphism $\theta _{\mathscr {G}, \mathscr {G}'}$ is a homotopy equivalence. Let us first regard the functor $\mathscr {G}'$ as fixed. Let $\widehat{\operatorname{\mathcal{C}}}'$ denote the full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by those objects $\mathscr {G}$ for which $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence. For each $K \in \mathbb {K}$, our assumption that $F$ preserves $K$-indexed colimits guarantees that the functor $\mathscr {G} \mapsto \theta _{ \mathscr {G}, \mathscr {G}' }$ preserves $K^{\operatorname{op}}$-indexed limits (Proposition 7.4.5.16). Consequently, the full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ is closed under the formation of $K$-indexed colimits. It will therefore suffice to show that $\theta _{ \mathscr {G}, \mathscr {G}' }$ is a homotopy equivalence in the special case where $\mathscr {G} = h_{C}$ is the functor represented by some object $C \in \operatorname{\mathcal{C}}$.
Let us now regard $\mathscr {G} = h_ C$ as fixed. Combining Example 8.4.6.2 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 $K$-indexed colimits, for each $K \in \mathbb {K}$. Invoking Corollary 8.4.3.9 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
\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,C') \ar [dl] \ar [dr] & \\ \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{C}}} }( \mathscr {G}, \mathscr {G}' ) \ar [rr]^{ \theta _{\mathscr {G}, \mathscr {G}'} } & & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(\mathscr {G}), F( \mathscr {G}') ), } \]
where the left vertical map is a homotopy equivalence by virtue of Yoneda's lemma (Theorem 8.3.3.13) and the right vertical map is a homotopy equivalence by virtue of assumption $(1)$. It follows that lower horizontal map is also a homotopy equivalence.
$\square$