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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