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)