John wrote [here](https://forum.azimuthproject.org/discussion/comment/20708/#Comment_20708):

> **Definition 12\\({}^\prime\\).** If \\(C\\) and \\(D\\) are monoidal categories, a monoidal functor \\(F \colon C \to D\\) is a **monoidal equivalence** if there is a monoidal functor \\(G \colon D \to C\\) such that there exist monoidal natural isomorphisms \\(\alpha \colon 1_C \Rightarrow FG \\), \\(\beta \colon GF \Rightarrow 1_D\\).

>3) However, there's a wonderful theorem that if we have \\(\alpha\\) and \\(\beta\\) as above, we can always _improve_ them in a systematic way to get new ones that _do_ satisfy the snake equations! Then we say we have an **adjoint equivalence**, because then \\(F\\) and \\(G\\) are also adjoint functors.

I drew some diagrams for weak inverses and adjoints using the definitions above:

Weak inverses

![weak_inverse_cap_cup](http://aether.co.kr/images/weak_inverse_cap_cup.svg)

Adjoints

![weak_inverse_cap_cup](http://aether.co.kr/images/adjoint_cap_cup.svg)

For the adjoint equivalence, once we set the snakes equations, FGF=F and GFG=G, and you get the diagram on the right.

> **Definition 12\\({}^\prime\\).** If \\(C\\) and \\(D\\) are monoidal categories, a monoidal functor \\(F \colon C \to D\\) is a **monoidal equivalence** if there is a monoidal functor \\(G \colon D \to C\\) such that there exist monoidal natural isomorphisms \\(\alpha \colon 1_C \Rightarrow FG \\), \\(\beta \colon GF \Rightarrow 1_D\\).

>3) However, there's a wonderful theorem that if we have \\(\alpha\\) and \\(\beta\\) as above, we can always _improve_ them in a systematic way to get new ones that _do_ satisfy the snake equations! Then we say we have an **adjoint equivalence**, because then \\(F\\) and \\(G\\) are also adjoint functors.

I drew some diagrams for weak inverses and adjoints using the definitions above:

Weak inverses

![weak_inverse_cap_cup](http://aether.co.kr/images/weak_inverse_cap_cup.svg)

Adjoints

![weak_inverse_cap_cup](http://aether.co.kr/images/adjoint_cap_cup.svg)

For the adjoint equivalence, once we set the snakes equations, FGF=F and GFG=G, and you get the diagram on the right.