Lemma 6.15.4 (007P)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.15: Algebraic structures
Lemma 6.15.4 (cite)

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Lemma 6.15.4. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $A, B, C \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ . Let $f : A \to B$ and $g : C \to B$ be morphisms of $\mathcal{C}$ . If $F(g)$ is injective, and $\mathop{\mathrm{Im}}(F(f)) \subset \mathop{\mathrm{Im}}(F(g))$ , then $f$ factors as $f = g \circ t$ for some morphism $t : A \to C$ .

Proof. Consider $A \times _ B C$ . The assumptions imply that $F(A \times _ B C) = F(A) \times _{F(B)} F(C) = F(A)$ . Hence $A = A \times _ B C$ because $F$ reflects isomorphisms. The result follows. $\square$

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