Lemma 47.2.2. Let $\mathcal{A}$ be an abelian category.

1. If $A \subset B$ and $B \subset C$ are essential extensions, then $A \subset C$ is an essential extension.

2. If $A \subset B$ is an essential extension and $C \subset B$ is a subobject, then $A \cap C \subset C$ is an essential extension.

3. If $A \to B$ and $B \to C$ are essential surjections, then $A \to C$ is an essential surjection.

4. Given an essential surjection $f : A \to B$ and a surjection $A \to C$ with kernel $K$, the morphism $C \to B/f(K)$ is an essential surjection.

Proof. Omitted. $\square$

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