Corollary

Let $(P, p)$ and $(P', p')$ be two projective resolutions of M. There exists a morphism, and only one up to homotopy, $\alpha : P' \to P$ such that $p \circ \alpha = p'$.

In fact, there exists a morphism $\alpha : P' \to P$ (resp. $\beta : P \to P'$) such that $p \circ \alpha = p'$ (resp. $p' \circ \beta = p$). Since $p \circ \alpha \circ \beta = p$ (resp. $p' \circ \beta \circ \alpha = p'$), $\alpha \circ \beta$ is homotopic to $1_P$ (resp. $\beta \circ \alpha$ is homotopic to $1_{P'}$).

PROPOSITION 3 bis. — Let $g : N \to N'$ be a homomorphism of a-modules, $e' : N' \to E'$ a morphism from $N'$ into a left zero and injective complex $E'$, and $e : N \to E$ a right resolution of N. There exists a morphism of complexes $\tilde{g} : E \to E'$, and only one up to homotopy, such that $\tilde{g} \circ e = e' \circ g$.

This is proved like Prop. 3 with the aid of Prop. 1 b).