First hypercohomology spectral sequence

In $C^{p,q}=F(I^{p,q})$ filter the signed total complex by the original complex degree $p$. Its graded column has differential $(-1{)}^{p}F(v)$. The kernel and image of this signed differential equal those of $F(v)$, so the canonical cohomology quotient is $H^q(F(I^{p,\bullet }))=R^{q}F(K^p)$ by the supplied injective resolution of $K^p$. The differential to the next column is induced by $F(h)$ and hence is $R^{q}F(d_K^p)$.

Apply finite-diagonal convergence. Its target is $H^{p+q}(\mathrm{Tot}(FI))={\mathbb{R}}_I^{p+q}F(K)$, and the filtration is the image of the cohomology of the subcomplex with columns of degree at least $p$. In total degree $n$ the filtration is all the target at $p=b$ and zero at $p=n+1$; below $n=b$ the target vanishes. Translation uses normalized total degree $n-b$.

For a map of complexes take the comparison of F3, which preserves columns. Its vertical homotopy makes the $E_1$ map independent of the lift, and its total homotopy makes the target map independent as well. Identity and composite comparisons prove naturality. This use requires exactly DC or the supplied comparisons stated above; constructing the sequence for the fixed bicomplex uses no choice. Zero complexes, zero columns and the sole bottom bidegree satisfy the same formulas.