>But now suppose we have two functors \$$F \colon \mathcal{A} \to \mathcal{B}\$$ and \$$G \colon \mathcal{B} \to \mathcal{A} \$$ with natural isomorphisms
>$\alpha \colon G F \Rightarrow 1_{\mathcal{A}} , \qquad \beta \colon F G \Rightarrow 1_{\mathcal{B}}$
>Then we say \$$F\$$ and \$$G\$$ are **weak inverses**, and the categories \$$\mathcal{A}\$$ and \$$\mathcal{B}\$$ are **equivalent**.

Setting it up like this gives a whole spectrum of associations versus the black/white version when you use an equal sign. Totally makes sense now why we use weak inverses or adjoints instead of inverses. You are ignoring a lot of information if you start with an equal sign!

The string diagrams for these illustrates the difference very well imo.

![weak equivalence](http://aether.co.kr/images/weak_equivalence.svg)