I think this must be well-known, but I can't find references, so my apologies from the very beginning.
Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ${\tt Vect}_{\mathbb K}$ of vector spaces over a given field $\mathbb K$ with a system of morphisms in this category (multiplication, unit, comultiplication and counit) such that the multiplication and the unit turn $B$ into an associative algebra, the comultiplication and the counit turn $B$ into a coassociative coalgebra, and a system of commutative diagrams is provided for the compatibility. These diagrams can be understood as the requirement that the comultiplication and the couint are morphisms in the category ${\tt Alg}_{\mathbb K}$ of algebras (or, equivalently the multiplication and the unit are morphisms in the category ${\tt Coalg}_{\mathbb K}$ of coalgebras). So the whole definition can be understood as follows:
a bialgebra is exactly a comonoid (=a coassociative coalgebra) in the category ${\tt Alg}_{\mathbb K}$ of associative algebras over $\mathbb K$ (or, equivalently, a monoid (=an associative algebra) in the category ${\tt Coalg}_{\mathbb K}$ of coassociative coalgebras over $\mathbb K$).
If we try to define Hopf algebras in the same way, then we face an obvious difficulty: the diagram for antipode can't be immediately interpreted as a diagram in the category ${\tt Alg}_{\mathbb K}$ of associative algebras.
I wonder if there exists a trick for overcoming this? Is it possible that the diagram for antipode can be changed into equivalent diagrams in ${\tt Alg}_{\mathbb K}$ (so that we obtain a correct definition of Hopf algebras as objects in ${\tt Alg}_{\mathbb K}$)?
