Let $\mathcal{C}$ and $\mathcal{D}$ be two equivalent categories and $F:\mathcal{C}\to \mathcal{D}$ be a functor that yields an equivalence of categories, with an inverse functor $G:\mathcal{D}\to \mathcal{C}$.

Now let us equip $\mathcal{D}$ with a tensor product $\otimes:\mathcal{D}\times\mathcal{D}\to \mathcal{D}$ and fix a unit object $I\in Ob(\mathcal{D})$, such that $(\mathcal{D},\otimes, I)$ forms a strict monoidal category.

Then the question is: can we naturally transfer the tensor product $\otimes$ and the unit object $I$ to corresponding structures on $\mathcal{C}$ such that $\mathcal{C}$ with the tranferred structures forms a monoidal category?