I am new at category theory and I haven't get the definition of adjoint functor. I have seen the another definition of natural transformation much similar to the definition of homotopy.
Other question in here gives the correlation between the basic concepts between topology and category, and it says that the adjoint functor is related to the homotopy equivalence. Thus I guess there is a definition of adjoint functor similar to the definition of homotopy equivalence, but I didn't get it.
Counit-unit adjunction seems to give most close definition of adjoint functor I wanted, but I fail to find the direct relation between the counit-unit adjunction and the homotopy equivalence.
So my question is: there is a definition of adjoint functor resembles to the definition of homotopy equivalence, as like the question in MathOverflow I mentioned as a link? I would appreciate any help.