  This paper is a fundamental study of comodules and contramodules over a
comonoid in a symmetric closed monoidal category. We study both algebraic and
homotopical aspects of them. Algebraically, we enrich the comodule and
contramodule categories over the original category, construct enriched functors
between them and enriched adjunctions between the functors. Homotopically, for
simplicial sets and topological spaces, we investigate the categories of
comodules and contramodules and the relations between them.
