Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of Higher Algebra that by identifying $C$ with a functor $N(Fin_\ast)\to Cat_\infty$ and composing with the involution $R:Cat_\infty\to Cat_\infty$, $C\mapsto C^{op}$, we can induce a symmetric monoidal structure on $C^{op}$ induced from that of $C$ which is unique up to contractible ambiguity (I'm fudging some details here, like the fact that this involution is actually defined on a different but equivalent category).

My question is the following: Let $C$ be a symmetric monoidal category. I want to consider the category of commutative algebras in $C^{op}$ (using the induced symmetric monoidal structure described above), $CAlg(C^{op})$. This category admits a symmetric monoidal structure of its own (see Remark 3.2.4.4 of Lurie's Higher Algebra). Thus, we can again take opposites to obtain a symmetric monoidal category $CoCAlg(C)\equiv(CAlg(C^{op})^{op})$. Let $\star$ denote the finally induced symmetric monoidal structure on $CoCAlg(C)$. Is it true that for $X$ and $Y$ objects of $CoCAlg(C)$, $X\star Y\simeq X\otimes Y$ where $\otimes$ is the original symmetric monoidal structure on $C$? If so, is there an easy way to prove this? It seems like a pretty reasonable thing to expect.

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