ScalingStacks

Proof. By A.2.4.2.7, the opposite of a symmetric monoidal $\infty$-category acquires a symmetric monoidal structure, which is Cartesian if and only if the original symmetric monoidal $\infty$-category is coCartesian. Hence, it is enough to prove (2). The unit object $1\in\mathcal{D}$ has a unique map from the initial object $\varnothing\to 1$. Since $f_{\alpha}$ is both symmetric monoidal and preserves finite coproducts, $f_{\alpha}\left(\varnothing\to 1\right)$ is the unique map from the initial object to the unit object of $\mathcal{C}_{\alpha}$, which is an equivalence by assumption. Since the collection of $f_{\alpha}$ is jointly conservative, it follows that the unit of $\mathcal{D}$ is initial in $\underline{\mathcal{D}}$ as well. Namely, $\mathcal{D}$ is unital as an $\infty$-operad. Using 2.2.3 we have a map of $\infty$-operads $G\colon\mathcal{D}\to\mathcal{D}_{\sqcup}$, which is an equivalence on the underlying $\infty$-categories. We need to show that this map is symmetric monoidal. Namely, that it maps coCartesian edges (over $\mathbf{Fin}_{*}$) to coCartesian edges. Since we already know that it is a map of $\infty$-operads and hence preserves inert morphisms, we only need to show that active coCartesian edges map to coCartesian edges. Using the Segal conditions, we are further reduced to considering only coCartesian lifts of the unique active morphism $\mu\colon\left\langle n\right\rangle\to\left\langle 1\right\rangle$. For every collection of objects $X_{1},X_{2},\dots,X_{n}\in\underline{\mathcal{D}}$, let

$$\mu_{\otimes}\colon X_{1}\oplus\cdots\oplus X_{n}\to X_{1}\otimes X_{2}\otimes\dots\otimes X_{n}$$

be a coCartesian lift of $\mu$ to $\mathcal{D}^{\otimes}$. Since $G$ is an equivalence on the underlying $\infty$-categories, $G\left(\mu_{\otimes}\right)$ can be considered as a map

$$X_{1}\oplus\cdots\oplus X_{n}\to X_{1}\otimes X_{2}\otimes\dots\otimes X_{n}$$

in $\mathcal{D}_{\sqcup}$. Now, let

$$\mu_{\sqcup}\colon X_{1}\oplus\cdots\oplus X_{n}\to X_{1}\sqcup X_{2}\sqcup\dots\sqcup X_{n}$$

be a coCartesian lift of $\mu$ to $\mathcal{D}^{\sqcup}$. There exists a unique (up to homotopy) map

$$g_{X_{1},\dots,X_{n}}\colon X_{1}\sqcup X_{2}\sqcup\dots\sqcup X_{n}\to X_{1}\otimes X_{2}\otimes\dots\otimes X_{n},$$

such that $G\left(\mu_{\otimes}\right)=g_{X_{1},\dots,X_{n}}\circ\mu_{\sqcup}$. We need to show that $g_{X_{1},\dots,X_{n}}$ is an equivalence in $\underline{\mathcal{D}}$ for all $X_{1},\dots,X_{n}\in\underline{\mathcal{D}}$. For every $\alpha$, we have a homotopy commutative diagram

in which the vertical and bottom maps are symmetric monoidal. It follows that $f_{\alpha}\left(g_{X_{1},\dots,X_{n}}\right)$ is an equivalence in $\mathcal{C}_{\alpha}$ for all $\alpha$. By joint conservativity, $g_{X_{1},\dots,X_{n}}$ is an equivalence as well. ∎