  We show that all coalgebras over the sphere spectrum are cocommutative in the
category of symmetric spectra, orthogonal spectra, $\Gamma$-spaces,
$\mathcal{W}$-spaces and EKMM $\mathbb{S}$-modules. Our result only applies to
these strict monoidal categories of spectra and does not apply to the
$\infty$-category setting.
