ScalingStacks

Proof. By A.3.2.4.4, the $\infty$-operad $\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)^{\otimes}\to\mathbf{Fin}_{*}$ is also a symmetric monoidal $\infty$-category and so we only need to show that the unit object of $\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)$ is initial. Since $\mathcal{Q}$ is unital, the canonical map $\mathcal{Q}\to\mathbb{E}_{0}\otimes\mathcal{Q}$ is an equivalence of $\infty$-operads (by A.2.3.1.9) and therefore the forgetful functor

$$\underline{\operatorname{Alg}}_{\mathbb{E}_{0}\otimes\mathcal{Q}}\left(\mathcal{C}\right)\simeq\underline{\operatorname{Alg}}_{\mathbb{E}_{0}}\left(\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)\right)\to\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)$$

is an equivalence of $\infty$-categories. On the other hand, by A.2.1.3.10 we have

$$\underline{\operatorname{Alg}}_{\mathbb{E}_{0}}\left(\operatorname{Alg}_{\mathcal{Q}}\left(\mathcal{C}\right)\right)\simeq\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)_{1/},$$

where $1\in\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)$ is the unit object and the projection $\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)_{1/}\to\underline{\operatorname{Alg}}_{\mathcal{Q}}\left(\mathcal{C}\right)$ is an equivalence of $\infty$-categories if and only if $1$ is initial (T.1.2.12.5). ∎