Proof of Proposition 9.7.1.6.
Since $\kappa < \lambda $ and $\lambda $ has exponential cofinality $\geq \kappa $, the collection of isomorphism classes of $\kappa $-small simplicial sets $K$ is $\lambda $-small (Proposition 4.9.4.20). Moreover, if the simplicial set $K$ is fixed, then the collection of isomorphism classes of diagrams $K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is $\lambda $-small (Remark 4.9.5.13). We may therefore assume without loss of generality that the index set $J$ is $\lambda $-small.
Let us say that an object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ is good if, for every object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, the morphism space $\operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) }( \mathscr {F}, \mathscr {G} )$ is essentially $\lambda $-small. Since $\operatorname{\mathcal{C}}$ is locally $\lambda $-small, every representable functor $\operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is good (see Proposition 8.3.1.3). Using Variant 7.4.1.15, we see that the collection of good objects is closed under $\kappa $-small colimits. For each $j \in J$, $q_{j}$ denote the restriction $\overline{q}_ j|_{ K_{j} }$ and let $\mathscr {F}_ j$ be a colimit of the diagram $(h \circ q_ j): K_{j} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$. Since the simplicial set $K_ j$ is $\kappa $-small, it follows that $\mathscr {F}_ j$ is both good and $(\kappa ,\lambda )$-compact. Let $Y_ j \in \operatorname{\mathcal{C}}$ denote the value of $\overline{q}_{j}$ at the cone point of $K_ j^{\triangleright }$, so that $h \circ \overline{q}_ j$ determines a morphism $w_{j}: \mathscr {F}_ j \rightarrow h_{ Y_ j }$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$.
Set $W = \{ w_ j \} _{j \in J}$. For every $\lambda $-cocomplete $\infty $-category $\operatorname{\mathcal{D}}$, let $\operatorname{Fun}_{Q}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ spanned by those functors $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which carry each $\overline{q}_ j$ to a colimit diagram in $\operatorname{\mathcal{D}}$, and $\operatorname{Fun}^{\lambda -\mathrm{cocont}}_{Q}( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }), \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }), \operatorname{\mathcal{D}})$ spanned by those $\lambda $-cocontinuous functors $F: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) \rightarrow \operatorname{\mathcal{D}}$ which carry each $h \circ \overline{q}_ j$ to a colimit diagram in $\operatorname{\mathcal{D}}$. Note that, if $F$ is $\lambda $-cocontinuous, then the latter condition is satisfied if and only if $F$ carries each $w_ j$ to an isomorphism in $\operatorname{\mathcal{D}}$. In the special case where $F$ is the functor represented by an object $\mathscr {G} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, we conclude that $\mathscr {G}$ belongs to $\overline{\operatorname{\mathcal{C}}}$ if and only if it is $W$-local. Assertion $(1)$ now follows from Proposition 9.6.3.10. To prove $(2)$, we note that there is a commutative diagram
\[ \xymatrix { \operatorname{Fun}^{ \lambda -\mathrm{cocont} }( \overline{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{D}}) \ar [r]^-{ \circ L} & \operatorname{Fun}^{ \lambda -\mathrm{cocont} }_ Q( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ), \operatorname{\mathcal{D}}) \ar [r]^-{\circ h} \ar [d] & \operatorname{Fun}_ Q( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \ar [d] \\ & \operatorname{Fun}^{ \lambda -\mathrm{cocont} }( \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ), \operatorname{\mathcal{D}}) \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \]
where the right side is a categorical pullback square and the left horizontal map is an equivalence of $\infty $-categories (Remark 9.6.3.11). It will therefore suffice to show that the horizontal map on the bottom right is an equivalence of $\infty $-categories, which follows from Theorem 8.4.3.2.
$\square$