Let $R$ be a unital ring. Let $\mathbf{A}_\bullet$ and $\mathbf{C}_\bullet$ be positive chain complexes of $R$-modules. If $\mathbf{A}_\bullet$ consists of flat $R$-modules then there is homology K"unneth spectral sequence $$E^2_{p,q}:=\bigoplus_{s+t=q}\mathrm{Tor}_p^R(H_s(\mathbf{A}_\bullet),H_t(\mathbf{C}_\bullet))\Rightarrow H_{p+q}(\mathbf{A}_\bullet\otimes_R\mathbf{C}_\bullet).$$

I am interested in a cohomological version, specifically, is the following true?

Suppose $\mathbf{C}_\bullet$ is a negative complex. If $\mathbf{A}_\bullet$ consists of projective $R$-modules, then there is a cohomology K"unneth spectral sequence $$E^2_{p,q}:=\bigoplus_{s+t=q}\mathrm{Ext}^p_R(H_s(\mathbf{A}_\bullet),H_t(\mathbf{C}_\bullet))\Rightarrow H_{p+q}(\mathrm{Hom}_R(\mathbf{A}_\bullet,\mathbf{C}_\bullet)).$$

A version of this appears in Rotman's introduction to homological algebra (first edition) but it does not appear in the second edition and I do not know of another reference.

If the ``theorem" is true, what is a reference for it?