Let $\Lambda$ be an associative ring with unity and let ${}_\Lambda\mathfrak M$ be a category of left unitary $\Lambda$-modules. The complete characterization of continuous additive co- and contravariant functors ${}_\Lambda\mathfrak M\to_\mathbb Z\mathfrak M$ is given. Such functors are either representable, or equivalent to a tenzor product, or the trivial functor. The class of categories, which are dual to ${}_\Lambda\mathfrak M$ and thefore equivalent to the category of compact right $\Lambda$-modules, is constructed by purely algebraic means. The canonical category is extracted from this class. The purely algebraic structure is constructed that is equivalent to the topology-algebraic structure of compact right $\Lambda$-module. Algebraic equivalents of connectivity and of complete inconnectivity are given.