Corollary 1

If $u$ is surjective, $K(u)$ is homotopic to zero (X, p. 34) as well as $K^A(u, C)$ and $K_A^*(u, C)$ for every complex $C$.

Indeed, there exists $x \in L$ such that $u(x) = 1$. Then $K(u)$ is homotopic to zero by the proposition 1, and therefore also $K^A(u, C)$ (X, p. 64, prop. 3) and $K_A^*(u, C)$ (X, p. 83, prop. 3).