A property of adjoint functors via universal morphisms

Question

Given a pair of adjoint functors $F\dashv G$. Let $\epsilon_X:FGX\to X$ be a universal arrow from $F$ to $X$ (it should exist due to the property of adjoint functors). Why does the following condition hold for any $f:X\to Y$: $$\epsilon_Y\circ FGf = f\circ \epsilon_X.$$ Or maybe there is some pair $\epsilon_X,\epsilon_Y$ for which this condition should hold. In this case is there a set of universal arrows (that consists of one universal arrow for each object) satisfiying the property that for any two objects $X,Y$ and for any morphism $f:X\to Y$, $e_X, e_Y$ belonging to this set satisfy $$\epsilon_Y\circ FGf = f\circ \epsilon_X.$$