  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.
