  We consider all Bott-Samelson varieties ${\rm BS}(s)$ for a fixed connected
semisimple complex algebraic group with maximal torus $T$ as the class of
objects of some category. The class of morphisms of this category is an
extension of the class of canonical (inserting the neutral element) morphisms
${\rm BS}(s)\hookrightarrow{\rm BS}(s')$, where $s$ is a subsequence of $s'$.
Every morphism of the new category induces a map between the $T$-fixed points
but not necessarily between the whole varieties.
  We construct a contravariant functor from this new category to the category
of graded $H^\bullet_T({\rm pt})$-modules coinciding on the objects with the
usual functor $H_T^\bullet$ of taking $T$-equivariant cohomologies.
  We also discuss the problem how to define a functor to the category of
$T$-spaces from a smaller subcategory. The exact answer is obtained for groups
whose root systems have simply laced irreducible components by explicitly
constructing morphisms between Bott-Samelson varieties (different from the
canonical ones).
