  By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita
context induces an equivalence between appropriate subcategories of the module
categories of the two rings in the Morita context. These are in fact categories
of firm modules for non-unital subrings. We apply this result to various Morita
contexts associated to a comodule $\Sigma$ of an $A$-coring $\cC$. This allows
to extend (weak and strong) structure theorems in the literature, in particular
beyond the cases when any of the coring $\cC$ or the comodule $\Sigma$ is
finitely generated and projective as an $A$-module. That is, we obtain
relations between the category of $\cC$-comodules and the category of firm
modules for a firm ring $R$, which is an ideal of the endomorphism algebra
$^\cC(\Sigma)$. For a firmly projective comodule of a coseparable coring we
prove a strong structure theorem assuming only surjectivity of the canonical
map.
