Proof.
We will show that $(1) \Leftrightarrow (2)$ and that $(4) \Rightarrow (2)$; the implications $(3) \Leftrightarrow (4)$ and $(2) \Rightarrow (4)$ then follow by applying the same arguments to the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. In what follows, we let $u$ and $v$ denote the cone points of the simplicial sets $K^{\triangleright }$ and $L^{\triangleleft }$, respectively.
We first show that $(1) \Leftrightarrow (2)$. Let $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(L^{\triangleleft }, \operatorname{\mathcal{C}})$ spanned by the limit diagrams in $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits $L$-indexed limits, the restriction functor
\[ \operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(L, \operatorname{\mathcal{C}}) \quad \quad \overline{u} \mapsto \overline{u}|_{L} \]
is a trivial Kan fibration, and therefore admits a section $s: \operatorname{Fun}(L, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$ which is an equivalence of $\infty $-categories. In particular, the $\infty $-category $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$ admits $K$-indexed colimits, so condition $(2)$ is equivalent to the requirement that the inclusion functor $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}(L^{\triangleleft }, \operatorname{\mathcal{C}})$ preserves $K$-indexed colimits. Using the criterion of Proposition 7.1.7.2, we see that this is equivalent to the requirement that the composite functor
\[ \operatorname{Fun}(L,\operatorname{\mathcal{C}}) \xrightarrow {s} \operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}(L^{\triangleleft }, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_{v} } \operatorname{\mathcal{C}} \]
preserves $K$-indexed colimits. Since this composition is isomorphic to the limit functor $\varprojlim : \operatorname{Fun}(L, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ (Remark 7.1.7.8), this is reformulation of condition $(1)$.
We now show that $(2) \Rightarrow (4)$. Suppose we are given a colimit diagram $f: K \rightarrow \operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$ which carries each vertex $k \in K$ to an object of $\operatorname{Fun}'( L^{\triangleleft }, \operatorname{\mathcal{C}})$; we wish to show that the colimit of $f$ has the same property. Choose an extension of $f$ to a colimit diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$, which we identify with a morphism of simplicial sets $\overline{F}: K^{\triangleright } \times L^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. The restriction $F|_{K^{\triangleright } \times L }$ can be identified with a diagram $g: L \rightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$. Our assumption that $\overline{f}$ is a colimit diagram guarantees that, for each vertex $\ell \in L$, the image $g_0( \ell )$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Since the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ admits $L$-indexed limits, we can extend $g$ to a limit diagram $\overline{g}: L^{\triangleleft } \rightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$. If condition $(4)$ is satisfied, then $\overline(v) \in \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ is also a colimit diagram in $\operatorname{\mathcal{C}}$. Let us identify $\overline{g}$ with a morphism of simplicial sets $\overline{G}: K^{\triangleright } \times L^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$, which we further identify with a diagram $\overline{f}': K^{\triangleright } \rightarrow \operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$. Since $K$-indexed colimits in $\operatorname{Fun}( L^{\triangleleft }, \operatorname{\mathcal{C}})$ are computed levelwise (Proposition 7.1.7.2), $\overline{f}'$ is a colimit diagram. Moreover, we have $\overline{f}'|_{K} = f$ by construction. We are therefore reduced to showing that $\overline{f}'(u) \in \operatorname{Fun}(L^{\triangleleft }, \operatorname{\mathcal{C}})$ is a limit diagram. This follows from our assumption that $\overline{g}$ is a limit diagram, since the evaluation functor $\operatorname{ev}_{u}: \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ preserves $L$-indexed limits.
$\square$