I was proving that:
(i) $F: \mathcal{C} \rightarrow \mathcal{D}$ is and equivalence of categories;
(ii) $F: \mathcal{C} \rightarrow \mathcal{D}$ is full, faithful and essentially surjective;
are equivalent propositions. Here, i'm assuming that equivalence was defined by the existence of a functor $G: \mathcal{D} \rightarrow \mathcal{C}$ such that there is natural isomorphisms $1_{\mathcal{C}} \simeq G \circ F$ and $1_{\mathcal{D}} \simeq F \circ G$.
Well, when i was doing the implication $(ii) \Rightarrow (i)$, i need to define the functor $G$ on objects, for this, i use the fact that $F$ is essentially surjective. Ok. Given an objet $X \in \mathcal{D}$ there exist $Y \in \mathcal{C}$ such that $F(Y) \simeq X$. So, i can choose a $Y$ in $\mathcal{C}$ wth this propriety and make $G(X) = Y$. But, this is not the axiom of choice being used? If the category is not small, how can a choose this object?
Assuming this i did all the rest of the proof, but this litle argument, i need to understand here.... There exist this kind of "choose function" for a propor class for exemple?
You are right: the implication (i) => (ii) holds in general, while the converse is equivalent to the axiom of choice. See some relevant discussion on the nLab.