A remark on monoidal structure and homological mirror symmetry

For a symplectic geometry $X$, suppose the (derived) Fukaya category $\mathrm{F}\mathrm{u}\mathrm{k}(X)$ of $X$ is equipped with a monoidal structure. Then its Balmer spectrum recovers a mirror $Y$ of $X$ if there exists homological mirror symmetry $\mathrm{F}\mathrm{u}\mathrm{k}(X)\cong D^b\mathrm{c}\mathrm{o}\mathrm{h}(Y)$ and the monoidal structure is the mirror of the standard one of $D^b\mathrm{c}\mathrm{o}\mathrm{h}(Y)$. In this short note, we fill one gap of this story in the literature: we show that the monoidal structure determines the homological mirror functor $\mathrm{F}\mathrm{u}\mathrm{k}(X)\to D^b\mathrm{c}\mathrm{o}\mathrm{h}(Y)$.