Proof.
Assertion $(1)$ follows from Proposition 7.1.8.2. We will prove $(2)$. Let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram having the property that, for each vertex $k \in K$, the object $F(k) \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact. If $K'$ is a simplicial set with a map $K' \rightarrow K$, we will say that $K'$ is good if $F|_{K'}$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{Fun}(K', \operatorname{\mathcal{C}})$. To complete the proof, it will suffice to show that if $K'$ is $\kappa $-small, then it is good.
Note that $\operatorname{Fun}(K',\operatorname{\mathcal{C}})$ can be realized as the inverse limit of a tower of $\infty $-categories
\[ \cdots \rightarrow \operatorname{Fun}( \operatorname{sk}_{2}(K'), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{sk}_{1}(K'), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{sk}_{0}(K'), \operatorname{\mathcal{C}}), \]
where the transition maps are isofibrations (Corollary 4.4.5.3). Each of the $\infty $-categories appearing in this tower is $(\kappa ,\lambda )$-cocomplete and each of the transition functors is $(\kappa ,\lambda )$-finitary. It will therefore suffice to show that each of the skeleta $\operatorname{sk}_{n}(K')$ is good: this is immediate if $\kappa = \aleph _0$ (the assumption that $K'$ is finite guarantees that it coincides with $\operatorname{sk}_{n}(K')$ for $n \gg 0$), and follows from Corollary 9.2.8.7 when $\kappa > \aleph _0$. We may therefore replace $K'$ by $\operatorname{sk}_{n}(K')$ and thereby reduce to the case where the simplicial set $K$ has dimension $\leq n$, for some integer $n \geq -1$.
We now proceed by induction on $n$. If $n = -1$, then the simplicial set $K'$ is empty and the result is clear. Otherwise, let $K'' = \operatorname{sk}_{n-1}(K')$ be the $(n-1)$-skeleton of $K$ and let $S$ be the collection of all nondegenerate $n$-simplices of $K'$ (which is $\kappa $-small, by virtue of our assumption that $K'$ is $\kappa $-small). Applying Proposition 1.1.4.12, we obtain a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( K', \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}( K'', \operatorname{\mathcal{C}}) \ar [d] \\ \prod _{\sigma \in S} \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \ar [r] & \prod _{\sigma \in S} \operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{C}}). } \]
Since the horizontal maps in this diagram are isofibrations (Corollary 4.4.5.3), it is also a categorical pullback square (Corollary 4.5.3.28). Moreover, each of the $\infty $-categories in the diagram is $(\kappa ,\lambda )$-cocomplete of the transition functors is $(\kappa ,\lambda )$-finitary. Combining our inductive hypothesis, Corollary 9.2.8.5, and Proposition 9.2.8.6, we can reduce to the case where $K' = \Delta ^ n$ is a standard simplex.
If $n = 0$, the desired result follows from our assumption that $F$ carries vertices of $K$ to $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$. If $n=1$, then we can identify $\operatorname{Fun}(K', \operatorname{\mathcal{C}})$ with the oriented fiber product $\operatorname{\mathcal{C}}\vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$, and the desired result follows from Proposition 9.2.8.3. We may therefore assume that $n \geq 2$. Let $\operatorname{Spine}[n] \subseteq \Delta ^ n$ be as in Example 1.5.7.7. The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne and therefore induces a trivial Kan fibration $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{C}})$. The desired result now follows from our inductive hypothesis, since the simplicial set $\operatorname{Spine}[n]$ has dimension $1$.
$\square$