Like Zhen Lin, I to doubt the existence of literature on this topic. However, let me provide the usual construction.
Equivalences of categories are defined such that they preserve every structure and property defined in category theory. Suppose $F^\prime: \mathcal{D}\to \mathcal{C}$ is a quasi-inverse to $F$. Define a monoidal structure on $\mathcal{D}$ as $X \otimes Y := F(G(X) \otimes G(Y))$ and $1_\mathcal{D} := F(1_\mathcal{C})$. You could check the associativity and the unit constraint, induced by $\mathcal{C}$. It can be observed that $F$ becomes a monoidal equivalence between $\mathcal{C}$ and $\mathcal{D}$.