  C-systems were defined by Cartmell as models of generalized algebraic
theories. B-systems were defined by Voevodsky in his quest to formulate and
prove an initiality conjecture for type theories. They play a crucial role in
Voevodsky's construction of a syntactic C-system from a term monad. In this
work, we construct an equivalence between the category of C-systems and the
category of B-systems, thus proving a conjecture by Voevodsky. We construct
this equivalence as the restriction of an equivalence between more general
structures, called CE-systems and E-systems, respectively. To this end, we
identify C-systems and B-systems as "stratified" CE-systems and E-systems,
respectively; that is, systems whose contexts are built iteratively via context
extension, starting from the empty context.
