Proof.
We first show that $(1)$ implies $(2)$. Assume that the functor $T$ preserves $\kappa $-small colimits and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ be a functor; we wish to show that the composite functor
\[ \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {T} \operatorname{\mathcal{D}} \]
is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. Lemma 8.4.3.5 guarantees that the $\infty $-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}$ is essentially $\kappa $-small. Since $T$ preserves $\kappa $-small colimits, it will suffice to show that the map $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )_{ / \mathscr {F}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is a colimit diagram (Remark 7.6.7.6), which follows from Corollary 8.4.2.2.
We now show that $(2)$ implies $(1)$. Assume that $T$ is left Kan extended from the $\infty $-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$; we wish to show that it preserves $\kappa $-small colimits. Choose a cardinal $\lambda $ such that $\operatorname{\mathcal{D}}$ is locally $\lambda $-small. Enlarging $\lambda $ if necessary, we may assume that it has exponential cofinality $\geq \kappa $ (Remark 4.7.3.19). By virtue of Proposition 7.4.5.16 (and Remark 7.4.5.18), it will suffice to show that for every representable functor $H: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$, the composition $H^{\operatorname{op}} \circ T$ preserves $\kappa $-small colimits. Since $H^{\operatorname{op}}$ preserves $\kappa $-small colimits (Proposition 7.4.5.16 and Remark 7.4.5.18), the functor $H^{\operatorname{op}} \circ T$ is left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Consequently, to show that $(2)$ implies $(1)$, we may replace $T$ by $H^{\operatorname{op}} \circ T$ and thereby reduce to the case where $\operatorname{\mathcal{D}}= (\operatorname{\mathcal{S}}^{< \lambda })^{\operatorname{op}}$, for some cardinal $\lambda $ of exponential cofinality $\geq \kappa $.
Let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$, and let $\mathscr {F}$ denote the composite functor
\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow { h_{\bullet }^{\operatorname{op}} } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\operatorname{op}} \xrightarrow { T^{\operatorname{op}} } \operatorname{\mathcal{S}}^{< \lambda }. \]
Using Remark 4.7.3.19 again, we can choose a cardinal $\lambda ' \geq \lambda $ of exponential cofinality $\geq \kappa $ such that $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$ is locally $\lambda '$-small. In what follows, we abuse notation by identifying $T$ with the composite functor $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \xrightarrow {T} ( \operatorname{\mathcal{S}}^{< \lambda } )^{\operatorname{op}} \hookrightarrow (\operatorname{\mathcal{S}}^{< \lambda '} )^{\operatorname{op}}$. Note that, since the inclusion $\operatorname{\mathcal{S}}^{< \lambda } \hookrightarrow \operatorname{\mathcal{S}}^{< \lambda '}$ preserves $\kappa $-small limits (see Variant 7.4.5.8), this composite functor is also left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$.
Let $H': \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}^{< \lambda '}$ be a functor represented by $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$, and let $U$ denote the composite functor
\[ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } ) \xrightarrow {H'^{\operatorname{op}} } (\operatorname{\mathcal{S}}^{< \lambda ' })^{\operatorname{op}}. \]
Applying Proposition 8.4.2.5, we see that the composition $U \circ h_{\bullet }$ is isomorphic to the functor $\mathscr {F} = T \circ h_{\bullet }$. Since the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$ is an equivalence of $\infty $-categories (Theorem 8.3.3.13), it follows that the functors $U$ and $T$ are isomorphic when restricted to $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Proposition 7.4.5.16 and Remark 7.4.5.18 guarantee that the functor $U$ preserves $\kappa $-small colimits. Invoking the implication $(1) \Rightarrow (2)$, we see that $U$ is left Kan extended from $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \kappa } )$. Applying the universal property of Kan extensions (Corollary 7.3.6.13), we deduce that the functor $T$ is isomorphic to $U$, and therefore also preserves $\kappa $-small colimits.
$\square$