Let $\mathcal{C}$ be a category with finite products. Then $\mathcal{C}$ is a braided monoidal category with the product as the monoidal product and terminal object as the monoidal unit, and braiding $\tau_{A,B}$ the unique isomorphism $A\times B \cong B\times A$ induced since both of sides are the product of $A$ and $B$. For every object $A$ in $\mathcal{C}$, $A$ has the unique structure of a comonoid with comultiplication given by the diagonal map, $\Delta$, and counit the unique map to the terminal object. Now suppose $A$ is a monoidal object with multiplication $\mu$. I'd like to show that the monoidal and comonoidal structures are compatible in the sense that $\Delta$ is a monoid morphism. This is part of the justification for calling Hopf algebras "group objects" in braided monoidal categories. I'm stuck trying to show that $\Delta\circ\mu = \mu\times\mu\circ(\text{id}\times\tau\times\text{id})\circ \Delta \times \Delta$.
Compatibility of monoid and comonoid structures when monoidal product is a product
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abstract-algebra
category-theory
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0By Yoneda Lemma (olr evalutation by representables) you can put $\mathcal{C}=Set$, now a monoid is a usual algebraic monoid, and is trivial see that $\Delta$ is a monoid morphism. – 2013-08-22