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 $\mathcal{D}$.
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