Comparing loose bimodules and double barrels using pseudo-models of enhanced sketches

(Pseudo) double categories have two sorts of morphisms: tight ones which compose strictly, and loose ones which compose up to coherent isomorphism. In this paper, we consider bimodules between double categories in the loose direction. We provide two formulation of this concept – first as pseudo-bimodules between pseudo-categories in the 2-category of categories, and second as double barrels generalizing Joyal’s definition of bimodules between categories as functors into the walking arrow – and prove these two formulations equivalent. In order to prove this equivalence, we define a notion of \emph{pseudo-model} of an enhanced sketch, which may be of independent interest. We then consider some double category theory unlocked by the theory of loose bimodules: loose adjunctions, and loose limits.