Corollary 9.5.4.2. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally cartesian closed (Definition 7.7.3.14).
Colimits in $\operatorname{\mathcal{C}}$ are universal: that is, every colimit diagram in $\operatorname{\mathcal{C}}$ is a universal colimit diagram.
Every small colimit diagram in $\operatorname{\mathcal{C}}$ is a universal colimit diagram.
Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 7.7.3.21 and the implication $(2) \Rightarrow (3)$ is trivial. We will complete the proof by showing that $(3)$ implies $(1)$. Assume that small colimits in $\operatorname{\mathcal{C}}$ are universal and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$; we wish to show that the pullback functor $f^{\ast }: \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}_{/X}$ admits a right adjoint (Proposition 7.7.3.19). Since the $\infty $-categories $\operatorname{\mathcal{C}}_{/X}$ and $\operatorname{\mathcal{C}}_{/Y}$ are presentable (Proposition 9.5.4.1), it will suffice to show that the functor $f^{\ast }$ preserves small colimits (Theorem 9.5.2.1). This is a reformulation of assumption $(3)$ (see Corollary 7.7.2.34). $\square$