ScalingStacks

Proof. For every $\left\langle n\right\rangle\in\mathbf{Fin}_{*}$, the restriction of $F^{\otimes}$ to the fiber over $\left\langle n\right\rangle$ is just $F^{n}\colon\mathcal{C}^{n}\to\mathcal{D}^{n}$, which is clearly a left adjoint. Hence, by A.7.3.2.7, the functor $F^{\otimes}$ is a left adjoint relative to $\mathbf{Fin}_{*}$. Let $G^{\otimes}$ be the right adjoint of $F^{\otimes}.$ Applying A.7.3.2.13, we obtain that $F^{\otimes}$ and $G^{\otimes}$ induce an adjunction:

$$\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)\leftrightarrows\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{D}\right).$$

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