We're given an adjunction $(F, U)$ where $F: C \to D, U : D \to C$ are the left and right adjoint functors.
We use the definition that the adjunction comes with a "unit of adjunction", or a natural map $\eta : 1_{C} \to U\circ F$ that satisfies: for any $f : c \to Ud$ in $C$ there is $g : Fc \to d$ such that $f = U(g) \circ \eta_c$.
This by definition induces a bijection of hom-sets: Now define $\phi : \text{Hom}_D(Fc, d) \simeq \text{Hom}_C(c, Ud)$.
My goal is to show that $\phi$, for fixed $d$ is natural in $c$. The proof in the book (Awodey) goes like this:
Let $h : c' \to c$ in $C$ and define $h^*$ to be composition with $h$ on the right: $h^*(a) = a \circ h$.
$$ h^*(\phi_{c, d}(f)) = (U(f) \circ \eta_c) \circ h \\ = U(f) \circ UF(h) \circ \eta_{c'} \\ = U(f \circ F(h)) \circ \eta_{c'} \\ = \phi_{c', d}(F(h)^*(f)) $$
But I'm not understanding how they arrived at the second line above starting from the first line.
The proof is on page 212 of "Category Theory" by Steve Awodey.