The Stacks project

Lemma 12.5.14. Let $\mathcal{A}$ be an abelian category.

  1. If $x \to y$ is surjective, then for every $z \to y$ the projection $x \times _ y z \to z$ is surjective.

  2. If $x \to y$ is injective, then for every $x \to z$ the morphism $z \to z \amalg _ x y$ is injective.

Proof. Immediately from Lemma 12.5.4 and Lemma 12.5.13. $\square$

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