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Like Zhen Lin, I too 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 $\mathcal{D}$.

Added: As Qiaochu says in the comments, "it shouldn't be true that you can transport a strict monoidal structure to another strict monoidal structure along an equivalence of categories". As Martin Brandenburg says here, strict monoidal categories are not definable in the (proper) language of categories. (Strict monoidal categories belong instead in set theory.)

Like Zhen Lin, I too 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 $\mathcal{D}$.

Added: As Qiaochu says in the comments, "it shouldn't be true that you can transport a strict monoidal structure to another strict monoidal structure along an equivalence of categories". As Martin Brandenburg says here, strict monoidal categories are not definable in the (proper) language of categories. (Strict monoidal categories belong instead in set theory.)

Like Zhen Lin, I too 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 $\mathcal{D}$.

Added: As Qiaochu says in the comments, "it shouldn't be true that you can transport a strict monoidal structure to another strict monoidal structure along an equivalence of categories". Strict monoidal categories are not definable in the (proper) language of categories. (Strict monoidal categories belong instead in set theory.)

Like Zhen Lin, I too 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 $\mathcal{D}$.

Added: As Qiaochu says in the comments, "it shouldn't be true that you can transport a strict monoidal structure to another strict monoidal structure along an equivalence of categories". As Martin Brandenburg says here, strict monoidal categories are not definable in the (proper) language of categories. (Strict monoidal categories belong instead in set theory.)

Like Zhen Lin, I too 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 $\mathcal{D}$.

Like Zhen Lin, I too 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 $\mathcal{D}$.

Added: As Qiaochu says in the comments, "it shouldn't be true that you can transport a strict monoidal structure to another strict monoidal structure along an equivalence of categories". Strict monoidal categories are not definable in the (proper) language of categories. (Strict monoidal categories belong instead in set theory.)