Recall the two definitions of equivalence of categories:
Definition 1. An equivalence of categories is a quadruplet $(F, G, \alpha, \beta)$, where $F : \mathbb{C} \to \mathbb{D}$ and $G : \mathbb{D} \to \mathbb{C}$ are functors and $\alpha : G F \Rightarrow \textrm{id}_\mathbb{C}$ and $\beta : F G \Rightarrow \textrm{id}_\mathbb{D}$ are natural isomorphisms.
Definition 2. A weak equivalence of categories is a fully faithful functor $F : \mathbb{C} \to \mathbb{D}$ that is also essentially surjective on objects, i.e. for each object $d$ in $\mathbb{D}$ there exists an object $c$ in $\mathbb{C}$ and an isomorphism $d \to F c$ in $\mathbb{D}$.
Assuming the axiom of choice, one can show that a weak equivalence $F : \mathbb{C} \to \mathbb{D}$ extends to an equivalence of categories $(F, G, \alpha, \beta)$. On the other hand, it is clear that this principle is equivalent to the axiom of choice: indeed, if we have some family of non-empty sets $( X_i : i \in I )$, we may form the category $\mathbb{C}$ whose object set is $\coprod_{i \in I} X_i$, such that there exists a unique arrow between two objects if and only if they come from the same set $X_i$; taking $\mathbb{D}$ to be the discrete category on the indexing set $I$, we have an evident projection functor $F : \mathbb{C} \to \mathbb{D}$, and by construction it is a weak equivalence; on the other hand, any functor $G : \mathbb{D} \to \mathbb{C}$ fitting into an equivalence $(F, G, \alpha, \beta)$ must yield a family $( x_i : i \in I )$ where $x_i = G i \in X_i$. Notice also that the categories $\mathbb{C}$ and $\mathbb{D}$ constructed here are groupoids (indeed, for that matter, setoids).
Question. Suppose $\mathbb{C}$ is a connected groupoid, i.e. between any two objects there exists at least one (iso)morphism between them. Then, if $\mathbb{D}$ is the full subcategory of $\mathbb{C}$ spanned by any single object, then the inclusion $G : \mathbb{D} \hookrightarrow \mathbb{C}$ is automatically a weak equivalence. Do I need the axiom of choice to extend $G$ to an equivalence of categories?