## On the two definitions of adjunctions

(Note that we write compositions from left to right and accordingly apply most maps on the right.)

Def.1
A functor   is called a left adjoint to a functor   if there is a bijection between the homsets:

natural in both and . In this case is called a right adjoint to , and we write .

Def.2
A functor is called a left adjoint to a functor if there are natural transformations and satisfying the zig-zag identities:

So, why are these two definitions equivalent?