>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)

>\[ \alpha \colon G F \Rightarrow 1_{\mathcal{A}} , \qquad \beta \colon F G \Rightarrow 1_{\mathcal{B}} \]

>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)