Proposition 12

Let E, F, G be three A-modules and $v : G \to F$, $u : E \to F$ homomorphisms. Suppose that E is finitely presented. For there to exist a homomorphism $w : E \to G$ such that $u$ factors into $E \xrightarrow{w} G \xrightarrow{v} F$, it is necessary and sufficient that, for all $m \in S_2$, there exist a homomorphism $w^m : E_m \to G$, such that $u_m : E_m \to F$, factors into $E \xrightarrow{w^m} G_m \xrightarrow{v_m} F$,

The existence of $w$ satisfying the above statement is equivalent to the following property: $u$ belongs to the image P of the mapping

Now, $(\operatorname{Hom}_A(E, F))_m$ (resp. $(\operatorname{Hom}_A(E, G))_m$) is canonically identified with $\operatorname{Hom}_{A_m}(E_m, F_m)$ (resp. $\operatorname{Hom}_{A_m}(E_m, G_m)$) (§ 2, no. 7, Proposition 19 (i)), the canonical image of $u$ in $(\operatorname{Hom}_A(E, F))_m$ is identified with $u_m$, $r_m$ is identified with $\operatorname{Hom}_{A_m}(1_{E_m}, v)$ and $P_m$ with the image of $r_m$. The proposition then follows from Corollary 1 to Theorem 1 applied to $\operatorname{Hom}_A(E, F)$ and its submodule P.