Completeness of the category of separable Chu spaces

In this paper we consider the category $\operatorname{Chu}_{Sep}(\mathbf{Set})$ of separable Chu spaces over the category $\mathbf{Set}$ of sets. The construction of the limit of an arbitrary functor into the category of Chu spaces over the category of sets is given when its images on objects are separable Chu spaces. The completeness of the category $\operatorname{Chu}_{Sep}(\mathbf{Set})$ is proved; constructions of equalizers, products and pullbacks in this category are given. It is shown that the colimits of separable Chu spaces are not always separable Chu spaces, but coproducts of separable Chu spaces in the category $\operatorname{Chu}_{Sep}(\mathbf{Set})$ exist for any separable Chu spaces.