Universal coefficient spectral sequence

Regard $C_q$ as a cochain complex in $(R\text{-}\mathrm{M}\mathrm{o}\mathrm{d}{)}^{\mathrm{o}\mathrm{p}}$, keeping the index $q$. The supplied projective Cartan–Eilenberg grid becomes injective Cartan–Eilenberg data there; cycles and boundaries exchange their kernel/cokernel descriptions and the homology object becomes the same cohomology object. The functor ${\mathrm{Hom}}_R(-,M)$ from this opposite category to abelian groups is additive and left exact: a map on a quotient is exactly a map vanishing on the submodule. On the homology object in degree $q$, F3 identifies its relative derived objects with ${\mathrm{Ext}}_{{\mathsf{P}}_q}^p(H_{q}C,M)$. Apply F2.

The horizontal-first sequence therefore has second page ${\mathrm{Ext}}_{{\mathsf{P}}_q}^p(H_{q}C,M)$. In the other sequence the first page is the relative Ext computed from the supplied term resolution of $C_q$. For $p>0$ it vanishes: because $C_q$ is projective, the first epimorphism in that resolution splits, its kernel is a projective summand, and the next epimorphism splits in turn. For each fixed degree this finite induction splits all short exact sequences needed to calculate that degree; Hom sends them to split exact sequences. In degree zero the kernel identification gives ${\mathrm{Hom}}_R(C_q,M)$. Thus the other sequence collapses after its cochain differential to the cohomology in F4.

The one-row collapse identifies the common total target with $H^n(\mathrm{Hom}(C,M))$. The second sequence retains its resolution-degree filtration, with endpoints zero and $n-b+1$ in degree $n\ge b$. F2 gives strong convergence and the stated bidegree and comparison naturality. At $n=b$ there is a single possible quotient; for $C=0$ with zero data everything is zero. The finite splitting argument in step 2.1 introduces no additional choice beyond the supplied data; global comparison conventions remain those of F2.