The smash product functor $X \wedge -: \mathcal{S}_* \to \mathcal{S}_*$ preserves colimits (being a left adjoint of the mapping space functor). By the adjoint functor theorem, it has a right adjoint, which is the pointed mapping space $\operatorname{Map}_*(X, -)$.

More generally, for any presentable $\infty$-category $\mathcal{C}$ tensored over $\mathcal{S}$, the tensor $X \otimes -: \mathcal{C} \to \mathcal{C}$ has a right adjoint (the cotensor/mapping object) whenever it preserves colimits.

For a ring homomorphism $\varphi: R \to S$, the extension of scalars functor $S \otimes_R^{\mathbf{L}} -: D(R) \to D(S)$ preserves all colimits. By the adjoint functor theorem, it has a right adjoint: the forgetful/restriction functor $\operatorname{Res}: D(S) \to D(R)$.

This gives the $\infty$-categorical version of the extension-restriction adjunction without needing to construct the right adjoint explicitly.

If $L: \mathcal{C} \to \mathcal{C}$ is an accessible localization functor (preserves colimits and is accessible), then $L$ has a right adjoint (the inclusion of local objects). This is immediate from the adjoint functor theorem.

Example: the truncation functor $\tau_{\leq n}: \mathcal{S} \to \mathcal{S}_{\leq n}$ preserves colimits and is accessible, so it has a right adjoint (the inclusion of $n$-truncated spaces into all spaces).

The inclusion $\operatorname{Shv}(\mathcal{C}) \hookrightarrow \operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathcal{S})$ preserves limits and is accessible. By the adjoint functor theorem, it has a left adjoint: the sheafification functor $L: \operatorname{Fun}(\mathcal{C}^{\mathrm{op}}, \mathcal{S}) \to \operatorname{Shv}(\mathcal{C})$.

For a functor $f: \mathcal{A} \to \mathcal{B}$ between small $\infty$-categories, the restriction functor $f^*: \operatorname{Fun}(\mathcal{B}, \mathcal{C}) \to \operatorname{Fun}(\mathcal{A}, \mathcal{C})$ preserves all limits and colimits. When $\mathcal{C}$ is presentable:

• $f^*$ has a left adjoint $f_!$ (left Kan extension) because $f^*$ preserves limits and is accessible.
• $f^*$ has a right adjoint $f_*$ (right Kan extension) because $f^*$ preserves colimits.

This gives the full adjunction $f_! \dashv f^* \dashv f_*$.

The forgetful functor $\Omega^\infty: \operatorname{Sp} \to \mathcal{S}_*$ (from spectra to pointed spaces) preserves limits and is accessible. By the adjoint functor theorem, it has a left adjoint $\Sigma^\infty: \mathcal{S}_* \to \operatorname{Sp}$ (the suspension spectrum functor).

The resulting adjunction $\Sigma^\infty \dashv \Omega^\infty$ is the fundamental link between unstable and stable homotopy theory.

In a closed monoidal presentable $\infty$-category $(\mathcal{C}, \otimes)$, for each object $X$, the tensor functor $X \otimes -: \mathcal{C} \to \mathcal{C}$ preserves colimits. By the adjoint functor theorem, it has a right adjoint $\underline{\operatorname{Hom}}(X, -): \mathcal{C} \to \mathcal{C}$, the internal hom.

This gives the tensor-hom adjunction $\operatorname{Map}(X \otimes Y, Z) \simeq \operatorname{Map}(Y, \underline{\operatorname{Hom}}(X, Z))$ automatically.

If a presentable $\infty$-category $\mathcal{C}$ has products and equalizers, then it has all limits. By the adjoint functor theorem, the diagonal functor $\Delta: \mathcal{C} \to \operatorname{Fun}(K, \mathcal{C})$ (for any $K$) has a right adjoint (the limit functor) if and only if it preserves colimits, which it does.

The adjoint functor theorem provides a recognition principle: an $\infty$-category $\mathcal{C}$ is presentable if and only if it is accessible and has all colimits. The theorem then guarantees that $\mathcal{C}$ also has all limits and that colimit-preserving functors from $\mathcal{C}$ always have right adjoints.