Steenrod-Cech homology-cohomology theories associated with bivariant functors

Let NG0 denote the category of all pointed numerically generated spaces and continuous maps preserving base-points. In [SYH], we described a passage from bivariant functors to generalized homology and cohomology theories. In this paper, we construct a bivariant functor such that the associated cohomology is the Cech cohomology and the homology is the Steenrod homology (at least for compact metric spaces).