category theory – Universal arrow provided by adjunction, erratum in MacLane?


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 $x$.