Lemma 1

a) If $C$ or $C'$ is bounded on the right, if $C$ is projective and if $H(C') = 0$, then $H(\mathrm{Homgr}_A(C, C')) = 0$. b) If $C$ or $C'$ is bounded on the left, if $C'$ is injective and if $H(C) = 0$, then $H(\mathrm{Homgr}_A(C, C')) = 0$.

Let $f \in Z_n(\mathrm{Homgr}_A(C, C'))$; $f$ is then a morphism of complexes from $C$ to $C'(n)$; in case a) (resp. b)), $f_m$ vanishes for $m$ sufficiently small (resp. sufficiently large). By X, p. 47, Proposition 1, $f$ is then homotopic to zero, hence belongs to $B_n(\mathrm{Homgr}_A(C, C'))$, which gives the conclusion.