Lemma 58.3.7. Let (\mathcal{C}, F) be a Galois category. Let X \to Y \in \text{Arrows}(\mathcal{C}). Then
F is faithful,
X \to Y is a monomorphism \Leftrightarrow F(X) \to F(Y) is injective,
X \to Y is an epimorphism \Leftrightarrow F(X) \to F(Y) is surjective,
an object A of \mathcal{C} is initial if and only if F(A) = \emptyset ,
an object Z of \mathcal{C} is final if and only if F(Z) is a singleton,
if X and Y are connected, then X \to Y is an epimorphism,
if X is connected and a, b : X \to Y are two morphisms then a = b as soon as F(a) and F(b) agree on one element of F(X),
if X = \coprod _{i = 1, \ldots , n} X_ i and Y = \coprod _{j = 1, \ldots , m} Y_ j where X_ i, Y_ j are connected, then there is map \alpha : \{ 1, \ldots , n\} \to \{ 1, \ldots , m\} such that X \to Y comes from a collection of morphisms X_ i \to Y_{\alpha (i)}.
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