How to prove equality of functions.

Question

Let $F:C\to D$ and $G:D\to C$ be functors such that $G\vdash F$.Let $x,y\in C$. Since,$G\vdash F$, we have that there is a natural bijection $j:D(Fx,Fy)\to C(x,GFy)$. Let $\eta:id_C\to GF$ be the unit of the adjunction. Note $\eta_{y_*}:C(x,y)\to C(x, GFy)$(this is post composition with $\eta_y$) is a function. Now consider the function $F_{x,y}:C(x,y)\to D(Fx,Fy)$. How would I prove that $\eta_{y_*}=j\circ F_{x,y}$?