Remark 7.7.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ admits finite products, then the forgetful functor $F: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ admits a right adjoint $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{/X}$ (Proposition 7.6.1.14). Moreover, the composition $(F \circ G): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is given on objects by the construction $C \mapsto C \times X$. Applying Corollary 6.2.4.2, we see that the following conditions are equivalent:
- $(1)$
For every object $Y \in \operatorname{\mathcal{C}}$, there exists an exponential $Y^{X} \in \operatorname{\mathcal{C}}$.
- $(2)$
The functor $(F \circ G): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ admits a right adjoint $E: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.
Moreover, if these conditions are satisfied, then the functor $E: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ is given on objects by the construction $Y \mapsto Y^{X}$.