Proof.
The implication $(1) \Rightarrow (2)$ is immediate from the definition. We next show that $(2) \Rightarrow (3)$. Assume that condition $(2)$ is satisfied. For each vertex $C \in \operatorname{\mathcal{C}}$, we can apply condition $(2)$ to the degenerate edge $\operatorname{id}_{C}$ to guarantee that the $\infty $-category $\Delta ^1 \times \operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-cocomplete. It follows immediately that the factor $\operatorname{\mathcal{E}}_{C}$ is also $(\kappa ,\lambda )$-cocomplete (see Example 7.1.3.11): that is, every $\lambda $-small $\kappa $-filtered diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{E}}_{C}$ can be extended to a colimit diagram $\overline{F}: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{C}$. We wish to show that in this case, $\overline{F}$ is an edgewise $U$-colimit diagram. To prove this, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^1$ and that $C \in \operatorname{\mathcal{C}}$ is the initial vertex. In this case, condition $(2)$ guarantees that the $\infty $-category $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete, so that $F$ extends to a colimit diagram $\overline{F}': \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{E}}$. Let us identify $\overline{F}$ and $\overline{F}'$ with objects $X,X' \in \operatorname{\mathcal{E}}_{F/}$. Our assumption that $\overline{F}'$ is a colimit diagram guarantees that $X'$ is an initial object of $\operatorname{\mathcal{E}}_{F/}$, so that there exists a morphism $X' \rightarrow X$. Then $\overline{F}'$ factors through the full subcategory $\operatorname{\mathcal{E}}_{C} \subseteq \operatorname{\mathcal{E}}$, and is therefore also a colimit diagram in $\operatorname{\mathcal{E}}_{C}$. It follows that $\overline{F}$ is isomorphic to $\overline{F}'$ (as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{\mathcal{E}}_{C} ) \subseteq \operatorname{Fun}( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{\mathcal{E}})$), and is therefore also a colimit diagram in $\operatorname{\mathcal{E}}$.
We now complete the proof by showing that $(3) \Rightarrow (1)$. Assume that condition $(3)$ is satisfied; we wish to show that for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete. Replacing $\operatorname{\mathcal{E}}$ by $\operatorname{\mathcal{E}}_{\sigma }$, we may assume without loss of generality that $\operatorname{\mathcal{C}}= \Delta ^ n$ is a standard simplex; in this case, we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}$ is $(\kappa ,\lambda )$-cocomplete. Let $\operatorname{\mathcal{K}}$ be an $\infty $-category which is $\lambda $-small and $\kappa $-filtered; we must show that every diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{E}}$ admits a colimit. For $0 \leq i \leq n$, let $\operatorname{\mathcal{K}}_{i} \subseteq \operatorname{\mathcal{K}}$ be the full subcategory spanned by those objects $K$ satisfying $(U \circ F)(K) = i$. Since $\operatorname{\mathcal{K}}$ is nonempty, there is some largest integer $0 \leq i \leq n$ for which $\operatorname{\mathcal{K}}_{i}$ is nonempty. Applying Corollary 9.1.5.17, we see that $\operatorname{\mathcal{K}}_ i$ is $\kappa $-filtered and the inclusion $\operatorname{\mathcal{K}}_ i \hookrightarrow \operatorname{\mathcal{K}}$ is right cofinal. Using Corollary 7.2.2.10, we can replace $F$ by $F|_{ \operatorname{\mathcal{K}}_{i} }$ and thereby reduce to the case where the diagram $F$ takes values in the $\infty $-category $\operatorname{\mathcal{E}}_{i}$. In this case, assumption $(3)$ guarantees that $F$ can be extended to a colimit diagram $\overline{F}: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{E}}_{i}$ which is also a $U$-colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ (Proposition 7.1.7.14). Since $U \circ \overline{F}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}= \Delta ^ n$, it follows that $\overline{F}$ is also a colimit diagram in the $\infty $-category $\operatorname{\mathcal{E}}$ (Corollary 7.1.6.12).
$\square$