Given categories $\mathbf{C}$ and $\mathbf{D}$, an equivalence from $\mathbf{C}$ to $\mathbf{D}$ is a pair of functors $F : \mathbf{C} \rightarrow \mathbf{D}$ and $G : \mathbf{D} \rightarrow \mathbf{C}$ together with natural isomorphisms $$\eta : FG \rightarrow \mathrm{id}_\mathbf{C} : \mathbf{C} \rightarrow \mathbf{C}$$ $$\varepsilon : \mathrm{id}_\mathbf{D} \rightarrow GF: \mathbf{D} \rightarrow \mathbf{D}$$
Honestly, I don't understand why these conditions are sufficient. Shouldn't we add some further requirements, like the assumption that $$\eta F \star F \varepsilon = \mathrm{id}_\mathrm{F} : F \rightarrow F : \mathbf{C} \rightarrow \mathbf{D}$$
for example?
(I've been away from math for awhile and can't remember standard category theory notation, but hopefully the meaning of the expression $\eta F \star F \varepsilon$ is fairly clear. Please comment if you're unsure of what I'm asking.)