In 2024, Migdail and Wehrli proved that odd Khovanov homology is functorial with respect to link cobordism (up to sign). Unlike Khovanov's proof that the original theory is functorial, Migdail-Wehrli's is interesting in that it does not depend on any of the recent extensions of odd Khovanov homology to tangles. In recent ongoing work, we adapt Khovanov's original argument to one of these tangle theories and get a proof that Naisse-Putyra's odd tangle theory is functorial with respect to tangle cobordism (up to unit). This approach motivates a few novel constructions, including a new generalization of Hochschild (co)homology.