Adjunctions via Universal Arrows

Question

Adjunctions via Universal Arrows: Understanding a Proof.

I got stuck here while reading this question.

$g=(g^♭)^♯$

How can I derive this equation?

Answer 1

For any morphism $f : A \to UY$, the adjunct $f^\sharp$ is defined to be the unique morphism $f^\sharp : FA \to Y$ such that $U(f^\sharp) \circ \eta_A = f$.

If $g: FA \to Y$ then $g^\flat: A \to UY$ is defined to be $Ug \circ \eta_A$. Then $(g^\flat)^\sharp$ is the unique morphism such that $U((g^\flat)^\sharp) \circ \eta_A = g^\flat$. Since $Ug\circ \eta_A = g^\flat$, by uniqueness we have $g = (g^\flat)^\sharp$.