In *Categories for the Working Mathematician*, p.81.

Every adjunction yields a universal arrow. Specifically, set $a = Fx$ in

$A(Fx,a) cong X(x, Ga)$ (1). The left hand hom-set of (1) then contains

the identity $1: Fx to Fx$; call its $varphi$-image $eta_x$. By Yoneda’s

Proposition III.2.1, this $eta_x$ is a universal arrow $eta_x: x to

> GFx$, $eta_x = varphi(1_{Fx})$, from $x in X$ to $G$.

Isn’t $eta_x$ a universal arrow from $Fx$ to $G$ ? As a first hint, $G$ is a functor $A to X$, so it can not be applied to $x$.