Lemma 22.27.5. In Situation 22.27.2 given a diagram
\xymatrix{x\ar[r]^ f\ar[d]_ a & y\ar[d]^ b\\ z\ar[r]^ g & w}
in \text{Comp}(\mathcal{A}) commuting up to homotopy. Then
If f is an admissible monomorphism, then b is homotopic to a morphism b' which makes the diagram commute.
If g is an admissible epimorphism, then a is homotopic to a morphism a' which makes the diagram commute.
Proof.
To prove (1), observe that the hypothesis implies that there is some h\in \mathop{\mathrm{Hom}}\nolimits _{\mathcal{A}}(x,w) of degree -1 such that bf-ga=d(h). Since f is an admissible monomorphism, there is a morphism \pi : y \to x in the category \mathcal{A} of degree 0. Let b' = b - d(h\pi ). Then
\begin{align*} b'f = bf - d(h\pi )f = & bf - d(h\pi f) \quad (\text{since }d(f) = 0) \\ = & bf-d(h) \\ = & ga \end{align*}
as desired. The proof for (2) is omitted.
\square
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