I was reading about equivalence of categories. So I came across two definitions and they both do not seem equivalent.
The first definition is the one given on Wikipedia: Given two categories $C$ and $D$, an equivalence of categories consists of a functor $F$ : $C$ → $D$, a functor $G$ : $D$ → $C$, and two natural isomorphisms $\epsilon: FG→I_D$ and $\eta : I_C→GF$.
The second definition is from the Grothendieck's translated paper here. The definition is on page four of the paper. In this definitioon he has an extra condition that the composite $F(A) \xrightarrow{F(\phi (A))}FGF(A) \xrightarrow{\psi^{-1}(F(A))} F(A)$ is an identity on $F(A)$.
So my question is are these two definitions equivalent. Thanks.