A spectral sequence for Tor

$\mathcal{A}, \mathcal{B}, \mathcal{C}$ are abelian categories, $F:\mathcal{A} \rightarrow \mathcal{B}$ and $G: \mathcal{B} \rightarrow \mathcal{C}$ are right exact functors. You want to compute right derived functors of $G\circ F$.

To apply's Grothendieck's spectral sequence you have to ensure that $F$ maps acyclic complexes in $\mathcal{A}$ to acyclic complexes in $\mathcal{B}$.

In your example $F(A) = A \otimes T$ and $G(B)= B \otimes C$, if $T$ is flat then $F$ takes acyclic complexes to acyclic ones but if $T$ is not flat then I do not know of a general characterization in that case.

a way around would be to extend your underlying category to complexes of modules. Then you can replace $T$ by a flat/free resolution. In that case Grothendieck's machinery will work but it will give hyper-tor instead of tor.