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 F:\ct A\to\ct B  is called a left adjoint to a functor U:\ct B\to\ct A  if there is a bijection between the homsets:

    \[\ct B(A^F\ig B) \simeq \ct A(A\ig B^U)\]

natural in both A and B. In this case U is called a right adjoint to F, and we write F\adj U.

Def.2
A functor F:\ct A\to\ct B is called a left adjoint to a functor U:\ct B\to\ct A if there are natural transformations \eta:1_{\ct A}\tto FU and \eps:UF\tto 1_{\ct B} satisfying the zig-zag identities:

    \[\matrix{\eta F\cdot F\eps = 1_F &\sep U\eta\cdot \eps U=1_U}\]

So, why are these two definitions equivalent?
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