Tensor products do not preserve quasi-isomorphisms

Example 6.6.24. Let \(\Aa = \Ab \) and consider the quasi-isomorphism \(f\colon C_{\bullet } \to D_{\bullet }\) where \(C_{\bullet }\) is the complex \(\Z \xrightarrow {\cdot 2} \Z \) concentrated in degrees \(1\) and \(0\), and \(D_{\bullet } = \Z /2\) concentrated in degree \(0\). Then \(\Z /2 \otimes f\) is the map \[ (\Z /2 \xrightarrow {0} \Z /2) \,\longrightarrow \, (0 \to \Z /2), \] which is not a quasi-isomorphism: the left-hand side has \(H_1 = \Z /2\), while the right-hand side has \(H_1 = 0\).