Lemma 12.7.3. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor. For every pair of objects $A, B$ of $\mathcal{A}$ the functor $F$ induces an abelian group homomorphism

$\mathop{\mathrm{Ext}}\nolimits _\mathcal {A}(B, A) \longrightarrow \mathop{\mathrm{Ext}}\nolimits _\mathcal {B}(F(B), F(A))$

which maps the extension $E$ to $F(E)$.

Proof. Omitted. $\square$

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