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 $\Delta\circ\mu = \mu\times\mu\circ(\text{id}\times\tau\times\text{id})\circ \Delta \times \Delta$.
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This has very little to do with monoidal categories and much more to do with the properties of the cartesian product (and, very specifically, $\Delta$). Observe: $$\begin{split} \Delta(\mu(x, y)) & = (\mu(x, y), \mu(x, y)) \\ & = (\mu \times \mu) (x, y, x, y) \\ & = ((\mu \times \mu) \circ (\textrm{id} \times \tau \times \textrm{id})) (x, x, y, y) \\ & = ((\mu \times \mu) \circ (\textrm{id} \times \tau \times \textrm{id}) \circ (\Delta \times \Delta)) (x, y) \end{split}$$ You can interpret $x$ and $y$ as generalised elements $U \to A$. |
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