  We show that weak monoidal Quillen equivalences induce equivalences of
symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan
localization of the symmetric monoidal model categories. The result will induce
a Dold-Kan correspondence of coalgebras in $\infty$-categories. Moreover, it
shows that Shipley's zig-zag of Quillen equivalences lifts to an explicit
symmetric monoidal equivalence of $\infty$-categories for the stable Dold-Kan
correspondence. We study homotopy coherent coalgebras associated to a monoidal
monoidal category. We show examples when these coalgebras cannot be rigidified.
That is, their $\infty$-categories are not equivalent to the Dwyer-Kan
localizations of strict coalgebras in the usual monoidal model categories of
spectra and of connective discrete $R$-modules.
