# 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 $$x$$.