TITLE: Direct sum of Hopf algebras
QUESTION [13 upvotes]: I realise that this question might be rather basic but however I was unable to find the answer in any textbook nor manage to figure out the answer. The question is the following: given two Hopf algebras $H_1,H_2$ is there a canonical way to turn the direct sum $H_1 \oplus H_2$ into a Hopf algebra? I have a problem already at the level of bialgebras: the counit is always a algebra morphism (linear and multiplicative) but the set of all linear multiplicative maps $\omega: H_1 \oplus H_2 \to \mathbb{C}$ is a sum of such sets corresponding to $H_1$ nad $H_2$. So I don't see how to define (in a canonical way) the counit map $\varepsilon_{H_1 \oplus H_2}$.

REPLY [5 votes]: This is a positive version of the answer "no".
As already mentioned before, the disjoint union of groups is not a group, BUT it is a grupoid!
In a similar way, the direct sum of two Hopf algebras is not a Hopf algebra, but a weak Hopf algebra. There is no problem for multiplication and comultiplication, the problem is, as you saw, the counit, but for weak Hopf algebras is the counit axiom that is precisely different.
You can check in
https://arxiv.org/pdf/2008.00606.pdf
UNIVERSAL QUANTUM SEMIGROUPOIDS, HONGDI HUANG, CHELSEA WALTON, ELIZABETH WICKS, AND ROBERT WON
the definition of weak Hopf (Definition 2.1),  Lemma 2.24 is precisely your question.