Wikipedia says:
An adjunction between categories $C$ and $D$ is somewhat akin to a "weak form" of an equivalence between $C$ and $D$.
I have heard this idea before, e.g. from Qiaochu.
Can you give an extremely simple example or argument that clarifies the intuitive idea that an adjunction is a “weak form of equivalence”? I’ve only read some explanations that are technical enough that my intuition doesn’t get that idea.
As the comments have mentioned, there are a few ways to think of this:
Equivalences are Adjunctions, but not conversely.
The unit/counit are natural isomorphisms if the adjunction is an equivalence.
An adjunction is a specific type of bijection between hom sets, or one could say they have the same number of morphisms/size of morphism sets. An equivalence already has this, but the adjunction is weaker but is quite ambient in Mathematics.
Helpful examples would include the forgetful functor $F:\textbf{Grp}\rightarrow \textbf{Set}$, and the free group functor $G:\textbf{Set}\rightarrow \textbf{Grp}$.
Grp and Set are definitely not equivalent, but there is a one-to-one correspondence between morphism sets defined by the adjunction between $F$ and $G$.