Given a double category $\mathbb D$ such that $\mathbb D_0$ has pushouts, we characterize oplax/lax adjunctions between $\mathbb D$ and $Cospan(\mathbb D_0)$ for which the right adjoint is normal and restricts to the identity on $\mathbb D_0$, where $Cospan(\mathbb D_0)$ is the double category on $\mathbb D_0$ whose vertical morphisms are cospans. We show that such a pair exists if and only if $\mathbb D$ has companions, conjoints, and 1-cotabulators. The right adjoints are induced by the companions and conjoints, and the left adjoints by the 1-cotabulators. The notion of a 1-cotabulator is a common generalization of the symmetric algebra of a module and Artin-Wraith glueing of toposes, locales, and topological spaces.
Keywords: double category, lax functor, (co)span, (co)tabulator, companion, conjoint, symmetric algebra
2010 MSC: 18D05, 18A40, 18B25, 18B30, 06D22, 18D10, 15A78
Theory and Applications of Categories, Vol. 26, 2012, No. 26, pp 729-742.