Lemma 22.5.2. Let $(A, \text{d})$ be a differential graded algebra. Let $f, g : L \to M$ be homomorphisms of differential graded $A$-modules. Suppose given further homomorphisms $a : K \to L$, and $c : M \to N$. If $h : L \to M$ is an $A$-module map which defines a homotopy between $f$ and $g$, then $c \circ h \circ a$ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$.

Proof. Immediate from Homology, Lemma 12.13.7. $\square$

Comment #289 by arp on

Typo: In the statement of the lemma, the homotopy $h$ should be a map $L \rightarrow M$ not $M \rightarrow N$.

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