Definition of Galois objects in a category I am using the reference by Lenstra. In page 41, section 3.14, he explains the definition of a Galois object in a category; but in doing this, he uses the claim (I think) that $F(A)$ is a finite set for $A$ a connected object, and $F$ the fundamental functor. I'm not sure if I'm missing something obvious under the technicalities, or if this was a corollary of a proposition.
 A: Let me quote the definition of a Galois category in the notes you refer to (section 3.1): "Let $C$ be a category and $F$ a covariant functor from $C$ to the category $sets$ of finite sets."
A: From what I understand, $\mathbf{C}$ is supposed to be a Galois category, as stated in 3.11:

3.11 Proof of Theorem 3.5. Let $\mathbf{C}$ be a Galois category with fundamental functor $F$. We begin with the proof of Theorem 3.5. Without loss of generality we assume that $\mathbf{C}$ is small (3.4).

This is also supported by the fact that the author appeals to the axioms of Galois category right after the sentence that this question is about.

3.14 Galois objects.
Let $A$ be connected. Then $\# \mathrm{Aut}_{\mathbf{C}}(A) ≤ \# \mathrm{Mor}_{\mathbf{C}}(A, A) ≤ \# F(A)$, so $\mathrm{Aut}_{\mathbf{C}}(A)$ is finite. We call $A$ a Galois object if the quotient $A / \mathrm{Aut}_{\mathbf{C}}(A)$ (axiom G2) is the
terminal object $1$.
[…]

Going back to the definition of a Galois connection on page 33, we see that $F(A)$ is finite by definition.

3.1 Definition.
Let $\mathbf{C}$ be a category and $F$ a covariant functor from $\mathbf{C}$ to the category $\mathbf{sets}$ of finite sets. We say that $\mathbf{C}$ is a Galois category with fundamental functor $F$ if the following
six conditions are satisfied.
[…]

