  We study the derived invariance of the cohomology theories $Hoch^*$, $H^*$
and $HC^*$ associated to coalgebras over a field. We prove a theorem
characterizing derived equivalences. As particular cases, it describes the two
following situations: 1) $f:C\to D$ a quasi-isomorphism of differential graded
coalgebras, 2) the existence of a "cotilting" bicomodule ${}_CT_D$. In these
two cases we construct a derived-Morita equivalence context, and consequently
we obtain isomorphisms $Hoch^*(C)\cong\Hoch^*(D)$ and $H^*(C)\cong H^*(D)$.
Moreover, when we have a coassociative map inducing an isomorphism $H^*(C)\cong
H^*(D)$ (for example when there is a quasi-isomorphism $f:C\to D$), we prove
that $HC^*(C)\cong HC^*(D)$.
