Right adjoint $R \colon C' \to C$ and left adjoint $L$.

$R$ is fully faithful iff the counit $\eta \colon LR \to id_{C'}$ is an isomorphism. The proof is given below from the textbook (Categories and Sheaves) using the commutative diagram.

Proof:
The map $Hom_{C'}(Y,Y') \to Hom_{C}(RY,RY')$ is bijective iff the map $Hom_{C'}(Y,Y') \to Hom_{C}(LRY,Y')$. Therefore, $R$ is fully faithful iff the map $LRY \to Y$ is an isomorphism for all $Y$. 

I know that this question has been answered in [![Adjunction diagram][1]][1] https://math.stackexchange.com/questions/3360363/right-adjoint-is-fully-faithful-iff-the-counit-is-an-isomorphism-without-yoneda 

I understand that, $R$ fully faithful, taking $Y'$ to be $LRY$ we get that $\eta_Y$ has a left inverse $g \colon Y \to Y'$. It was not clear how is $g$ also a right inverse. I somewhat know the explanation has to be simple but I am being foolish about it.


  [1]: https://i.sstatic.net/iVWc5D1j.png