  After two papers on weak cubical categories and {\it collarable} cospans,
respectively, we put things together and construct a {\it weak} cubical
category of cubical {\it collared} cospans of topological spaces. We also build
a second structure, called a {\it quasi} cubical category, formed of arbitrary
cubical cospans concatenated by homotopy pushouts. This structure, simpler but
weaker, has {\it lax} identities. It contains a similar framework for
cobordisms of manifolds with corners and could therefore be the basis to extend
the study of TQFT's of Part II to higher cubical degree.
