Let $K$ be a field and $\mathrm{Hopf}^K_{E_\infty}$ the $\infty$-category of homotopy-coherent hopf algebras over $K$ that are coherently commutative and cocommutative. Precisely, $\mathrm{Hopf}^K_{E_\infty}$ is the $\infty$-category of abelian group objects (in the derived sense) in the $\infty$-category $\mathrm{Coalg}_{E_\infty}^K$ of $E_\infty$-coalgebras over $K$. Homology defines a functor $$H_*: D(K) \to \mathrm{grVect}_K $$ from the derived $\infty$-category of $K$ to the category of $\mathbb{Z}$-graded $K$-vector spaces. By the Künneth theorem homology $H_*: D(K) \to \mathrm{grVect}_K $ is symmetric monoidal and so induces a functor $$H_*:\mathrm{Hopf}^K_{E_\infty} \to \mathrm{grHopf}_K, $$ where $\mathrm{grHopf}_K$ is the category of commutative and cocommutative $\mathbb{Z}$-graded hopf algebras over $K.$
The category $\mathrm{grHopf}_K$ is abelian and so it makes sense to talk about exact sequences in $\mathrm{grHopf}_K$. Moreover it makes sense to talk about lim^1 of a tower in $\mathrm{grHopf}_K$. Given a tower $A_\bullet$ in $\mathrm{grHopf}_K$ is there for any $n \in \mathbb{N}$ a short exact sequence $$ 0 \to lim^1(H_{n+1}(A_\bullet)) \to H_{n}(holim(A_\bullet)) \to lim(H_{n}(A_\bullet)) \to 0 $$ in the abelian category $\mathrm{grHopf}_K$, where the (homotopy) inverse limits are taken in $\mathrm{Hopf}^K_{E_\infty}$ and $\mathrm{grHopf}_K$?