  In this work, we compute the topological coHochschild homology (coTHH) of
interesting coalgebras such as the Steenrod algebra spectrum. For this, we
start by extending the Hess-Shipley definition of coTHH to $\infty$-categories,
following the Nikolaus-Scholze approach to THH. Furthermore, we prove that
coTHH of what we call quasi-proper coalgebras can be obtained from THH via
Spanier-Whitehead duality which provides further insight into coTHH and its
relationship to THH.
