Multiplication of a Bialgebra Is a Coalgebra Hom
Let \(R\) be a commutative ring and \(A\) an \(R\)-bialgebra with multiplication \(\mu : A \otimes_R A \to A\), comultiplication \(\Delta : A \to A \otimes_R A\), counit \(\varepsilon : A \to R\), and unit \(\eta : R \to A\). Then \(\mu\) is an \(R\)-coalgebra homomorphism from \(A \otimes_R A\) (with its tensor-bialgebra structure) to \(A\). Concretely, \(\mu\) satisfies the two coalgebra-hom axioms:
\[ \varepsilon_A \circ \mu \;=\; \varepsilon_{A \otimes A}, \qquad (\mu \otimes \mu) \circ \Delta_{A \otimes A} \;=\; \Delta_A \circ \mu. \]
Proof
By definition of the bialgebra \(A\), the counit \(\varepsilon\) and the comultiplication \(\Delta\) are themselves \(R\)-algebra homomorphisms. Spelled out at the level of elements (using Sweedler notation):
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Counit axiom. For all \(a, b \in A\), \(\varepsilon(a \cdot b) = \varepsilon(a) \cdot \varepsilon(b)\) (since \(\varepsilon\) is an algebra hom). The right-hand side is exactly \(\varepsilon_{A \otimes A}(a \otimes b)\) (by the tensor product of coalgebras: \(\varepsilon_{A \otimes A} = m_R \circ (\varepsilon \otimes \varepsilon)\)). Hence \(\varepsilon_A \circ \mu = \varepsilon_{A \otimes A}\).
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Comultiplication axiom. For all \(a, b \in A\), \(\Delta(a \cdot b) = \Delta(a) \cdot \Delta(b)\) (since \(\Delta\) is an algebra hom), where the right-hand side is the algebra multiplication in \(A \otimes A\). Writing \(\Delta(a) = \sum a_{(1)} \otimes a_{(2)}\) and \(\Delta(b) = \sum b_{(1)} \otimes b_{(2)}\), this gives
\[ \Delta(a \cdot b) \;=\; \sum (a_{(1)} b_{(1)}) \otimes (a_{(2)} b_{(2)}) \;=\; (\mu \otimes \mu)\bigl(\Delta_{A \otimes A}(a \otimes b)\bigr), \]
where the last equality uses the tensor-bialgebra comultiplication \(\Delta_{A \otimes A}(a \otimes b) = \sum (a_{(1)} \otimes b_{(1)}) \otimes (a_{(2)} \otimes b_{(2)})\) (the "tensor of comuls plus swap of middle factors"). \(\square\)
Downstream use
The coalgebra-hom structure of \(\mu\) is the key ingredient in the
convolution-inverse uniqueness proof of antipode multiplicativity for
commutative Hopf algebras (antipode_multiplicativity_commutative_hopf).
It lets one pull back the antipode identities from
\(\operatorname{End}_R(A)\) to \(\operatorname{Hom}_R(A \otimes A, A)\)
via pre-composition by \(\mu\).