Spanier–Whitehead duality for topological coHochschild homology

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