Invariance properties of coHochschild homology

The notion of Hochschild homology of a dg algebra admits a natural dualization, the coHochschild homology of a dg coalgebra, introduced in arXiv:0711.1023 by Hess, Parent, and Scott as a tool to study free loop spaces. In this article we prove "agreement" for coHochschild homology, i.e., that the coHochschild homology of a dg coalgebra $C$ is isomorphic to the Hochschild homology of the dg category of appropriately compact $C$-comodules, from which Morita invariance of coHochschild homology follows. Generalizing the dg case, we define the topological coHochschild homology (coTHH) of coalgebra spectra, of which suspension spectra are the canonical examples, and show that coTHH of the suspension spectrum of a space $X$ is equivalent to the suspension spectrum of the free loop space on $X$, as long as $X$ is a nice enough space (for example, simply connected.) Based on this result and on a Quillen equivalence established by the authors in arXiv:1402.4719, we prove that "agreement" holds for coTHH as well.