We study the composition of modules between lax functors of weak double categories. We adapt the bicategorical notion of local cocompleteness to weak double categories, which the codomain of our lax functors will be assumed to satisfy. We introduce a notion of factorization of cells, which most weak double categories of interest possess, and which is sufficient to guarantee the strong representability of composites of modules between lax functors whose domain satisfies it.
Keywords: double category, lax functor, module, modulation, representability
2010 MSC: 18D05,18D25
Theory and Applications of Categories, Vol. 27, 2013, No. 16, pp 393-444.