A contractible two-term bimodule complex induces the zero tensor functor

The only nonzero differential of $F$ is the degree-zero bimodule map $d_F^{-1}={\mathrm{id}}_A$; every composite of two consecutive differentials is zero because the next differential is zero, so $F$ is a bounded complex supported at the endpoints $-1$ and $0$.

Define $k^0:F^0\to F^{-1}$ to be ${\mathrm{id}}_A$ and all other components to be zero; then $k^0{d}_F^{-1}=\mathrm{id}$ in degree $-1$ and $d_F^{-1}{k}^0=\mathrm{id}$ in degree $0$, hence $d_{F}k+k{d}_F={\mathrm{id}}_F$, with every component internal-degree preserving and bimodule-linear.

For any bounded graded left $A$-complex $X$, the signed totalization has $T^n=(A{\otimes }_A{X}^{n+1})\oplus (A{\otimes }_A{X}^n)$; on elementary tensors $(a\otimes x,b\otimes y)$, with $x\in X^{n+1}$ and $y\in X^n$, its differential is $D^n(a\otimes x,b\otimes y)=(-a\otimes d_{X}x,a\otimes x+b\otimes d_{X}y)$, where the first sign is $(-1{)}^{-1}$ and the second-factor signs are $(-1{)}^{-1}$ and $(-1{)}^0$ on the two rows, and the formula extends additively to each balanced total term.

Each nonzero term $A=A\{0\}$ is a degree-zero direct summand of itself and hence finite graded projective on the left by [L3]; as a right module it is projective because, viewed as a left $A^{op}$-module, any fixed surjection $q:E\twoheadrightarrow M$ and right-linear $f:A\to M$ admit $e$ with $q(e)=f(1)$, and $\tilde{f}(a)=ea$ is a right-linear lift by [L4], while zero terms are projective on both sides. Thus $F$ and the zero complex satisfy [L5] and [L8].

By [L2], $H^n(a\otimes x,b\otimes y)=(b\otimes y,0)$; then $D^{n-1}{H}^n(a\otimes x,b\otimes y)=(-b\otimes d_{X}y,b\otimes y)$ and $H^{n+1}{D}^n(a\otimes x,b\otimes y)=(a\otimes x+b\otimes d_{X}y,0)$, whose sum is $(a\otimes x,b\otimes y)$ because the mixed terms cancel, also in characteristic two. Thus $dH+Hd={\mathrm{id}}_T$.

Let $0$ be the zero bimodule complex and take the zero maps $u:F\to 0$, $v:0\to F$, the homotopy $-k$ for $vu-{\mathrm{id}}_F=-{\mathrm{id}}_F$, and the zero homotopy for $uv-{\mathrm{id}}_0=0$; [L9] gives natural isomorphisms of their tensor functors on the bounded projective homotopy category and on ordinary and graded bounded derived categories, while [L6] and [L7] identify the latter with derived tensor and $0{\otimes }_A-$ is zero.

For every chain map $g:X\to Y$, both composites in the naturality square for $H$ send $(a\otimes x,b\otimes y)$ to $(b\otimes g(y),0)$, so the contraction is natural on bounded complexes.

If $X$ is zero or has empty support, all terms and homotopy maps are zero; if $X$ is concentrated in one degree the same formula applies with missing rows zero; if $d_X=0$ the differential terms vanish but step 2.1 still gives $dH+Hd=\mathrm{id}$. If $X$ is supported in $[c,d]$, step 1.3 gives $T$ support $[c-1,d]$, and all terms and maps outside those bounded endpoints are zero. The contraction is explicit, and the projectivity argument in step 1.4 uses only one preimage for one fixed lifting square, so no Axiom of Choice is used; the example states no iff claim. [step 1.3, step 2.1, step 1.4, algebra] $\mathrm{\square }$