Hochschild (co)homology for odd Khovanov arc algebras

We extend Hochschild homology and cohomology to quasi-associative algebras, which were defined initially by Albuquerque and Majid and generalized by Naisse and Putyra via grading categories. As an application, we use our construction to give an alternative proof of up-to-sign functoriality for odd Khovanov homology, which was recently proven by Migdail and Wehrli. Our proof generalizes an argument of Khovanov: in doing so, we also give the first proof that the tangle theory of Naisse and Putyra is functorial with respect to tangle cobordisms, up to unit.