  The purpose of this paper is to give a pair of canonical extensions of
Mayer-Vietoris functors (e.g. homology or cohomology theories of spaces) to
cospans of spaces. One of the pair, called cospanical extension in this paper,
is closely related with the classical theory part of abelian
Dijkgraaf-Witten-Freed-Quinn model and bicommutative
Turaev-Viro-Barrett-Westbury model. We derive those extensions by revealing a
partial universality of a cospan category of CW-spaces which is a
homotopy-theoretic analogue of cobordism categories.
