Bialgebras and bialgebra maps

Fix a field $k$. A bialgebra over $k$ is a $k$-vector space $H$ with a unital associative $k$-algebra structure $(m,u)$ and a counital coassociative $k$-coalgebra structure $(\mathrm{\Delta },\epsilon )$ in the sense of Coalgebra and coalgebra map over a field, where $m:H\otimes H\to H$, $u:k\to H$, $\mathrm{\Delta }:H\to H\otimes H$, and $\epsilon :H\to k$. Explicitly, the algebra equations are $m(m\otimes \mathrm{i}\mathrm{d})=m(\mathrm{i}\mathrm{d}\otimes m)$ and $m(u\otimes \mathrm{i}\mathrm{d})=\mathrm{i}\mathrm{d}=m(\mathrm{i}\mathrm{d}\otimes u)$ under the canonical identifications; the coalgebra equations are those of Coalgebra and coalgebra map over a field.

Require $\mathrm{\Delta }:H\to H\otimes H$ and $\epsilon :H\to k$ to be unital algebra homomorphisms. The algebra on $H\otimes H$ is the one with $(a\otimes b)(c\otimes d)=ac\otimes bd$ and unit $1_H\otimes 1_H$; the algebra on $k$ is its field algebra. A bialgebra map $f:H\to K$ is a $k$-linear map preserving multiplication, unit, comultiplication, and counit: $f{m}_H=m_K(f\otimes f)$, $f{u}_H=u_K$, ${\mathrm{\Delta }}_{K}f=(f\otimes f){\mathrm{\Delta }}_H$, and ${\epsilon }_{K}f={\epsilon }_H$.