Let $F: C \rightarrow D,\ G: D \rightarrow C$ be an adjunction of functors. My goal is to construct an isomorphism
$$Hom_D(F(x), y) \simeq Hom_C(x, G(y))$$
So given a morphism $f(x): F(x)\rightarrow y$, I can form a morphism $g: x \rightarrow G(y)$ by
$$G(f)\circ e_x: x \rightarrow GF(x) \rightarrow G(y)$$
Conversely, given $g: x \rightarrow G(y)$, I can form some $f': F(x) \rightarrow y$, by
$$\epsilon_y \circ F(g): F(x) \rightarrow F(G(y)) \rightarrow y$$
I feel to see why $f = f'$, ie why these two mappings of morphisms I defined are inverses of each other. I'm looking for an answer that carefully spells out why this is the case.