Lemma 15.28.3. Let $\varphi : E \to R$ and $\varphi ' : E' \to R$ be $R$-module maps. Let $\psi : E \to E'$ be an $R$-module map such that $\varphi ' \circ \psi = \varphi $. Then $\psi $ induces a homomorphism of differential graded algebras $K_\bullet (\varphi ) \to K_\bullet (\varphi ')$.

**Proof.**
This is immediate from the definitions.
$\square$

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