Let $\mathbb{K}$ be a suitable field and $(\mathcal{Ch}_{\mathbb{K}},\otimes)$ the "standard" symmetric monoidal category of chain complexes over that field, with the symmetric tensor product, induced by the commutativity constraint $a\otimes b = (-1)^{|a||b|}b\otimes a$.
Considering graded vector spaces as chain complexes with trivial differential, the homology is an endofunctor $H: \mathcal{Ch}_{\mathbb{K}} \to \mathcal{Ch}_{\mathbb{K}}$ that maps any chain complex to its homology.
A key feature of $H$ is, that it is a strict symmetric monoidal functor. Now my question is: In what sense (if any) is $H$ a symmetric comonoidal functor?