Corollary 7.6.2.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
- $(1)$
For every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the composition functor
\[ \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y} \quad \quad e \mapsto (f \circ e) \]of Example 4.3.6.15 admits a right adjoint.
- $(2)$
For every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$, the composition functor
\[ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ X\} \rightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} \quad \quad e \mapsto (f \circ e) \]admits a right adjoint.
- $(3)$
The evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a locally cartesian fibration of $\infty $-categories.
- $(3')$
The evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a cartesian fibraton of $\infty $-categories.
- $(4)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits pullbacks.