In this talk, I present a recent result of Brooke Shipley and I which shows that homotopy coherent coalgebras in spectra cannot be rigidified, i.e. are not equivalent to strictly coassociative coalgebras. I will also present the main result of my PhD thesis which shows that the rigidification occurs for comodules in dg-context. If time permits, I will also mention current project with Jonathan Beardsley on certain comodules in spectra.