  Steinberg pro-groups are certain pro-groups used to analyze ordinary
Steinberg groups locally in Zariski topology. In this paper we show that
Steinberg pro-groups associated with general linear groups, odd unitary groups,
and Chevalley groups satisfy a Zariski cosheaf property as crossed pro-modules
over the base groups. Also, we prove an analogue of the standard commutator
formulae for relative Steinberg groups. As an application, we show that the
base groups over localized rings naturally act on the corresponding Steinberg
pro-groups.
