Lemma 14.27.1. Let $\mathcal{A}$ be an abelian category. Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{A}$. If $a$, $b$ are homotopic, then $s(a), s(b) : s(U) \to s(V)$, and $N(a), N(b) : N(U) \to N(V)$ are homotopic maps of chain complexes.

Proof. The part about $s(a)$ and $s(b)$ is clear from the calculation above the lemma. On the other hand, if follows from Lemma 14.23.6 that $N(a)$, $N(b)$ are compositions

$N(U) \to s(U) \to s(V) \to N(V)$

where we use $s(a)$, $s(b)$ in the middle. Hence the assertion follows from Homology, Lemma 12.12.1. $\square$

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