Let ($\mathcal{D}, \otimes, 0)$ be a symmetric monoidal category enriched over pointed spaces, and denote by $\mathsf{Top}_*^\mathcal{D}$ to the category of "$\mathcal{D}$-spaces", the category of enriched functors.
There is a closed symmetric monoidal structure on $\mathcal{D}$-spaces given by Day convolution (a left Kan extension): if $D \in \mathcal{D}$ and $X,Y$ are $\mathcal{D}$-spaces, then $$(X \wedge Y)_D := \int^{A, B \in \mathcal{D}} \mathrm{Hom}_\mathcal{D} (A \otimes B, D) \wedge X_D \wedge Y_B.$$
Let $R$ be a commutative monoid object on $\mathcal{D}$-spaces, so there is a commutative diagram saying that the multiplication map $R \wedge R \to R$ and $$R \wedge R \overset{\cong}{\to} R \wedge R \to R$$ (symmetry isomorphism and then multiplication) coincide.
Question. Is $R$ commutative if and only if the diagram
$$\require{AMScd} \begin{CD} R_D \wedge R_{D'} @>>> R_{D \otimes D'}\\ @VVV @VVV \\ R_{D'} \wedge R_D @>>> R_{D' \otimes D} \end{CD}$$
commutes? According to Schwede, it should be the case, but I have not been able to give a proof of this, linking this explicit description with the symmetry isomorphism of the commutative monoid object.