Functoriality of bornological groupoid convolution

We show that the complete bornological convolution algebras of Lie groupoids and convolution bimodules of groupoid bibundles define a monoidal functor from the 2-category of differentiable stacks to the Morita 2-category of complete bornological algebras. The convolution algebras are generally non-unital, but are shown to possess one-sided approximate units such that the multiplication operators Mackey converge in the functional bornology of endomorphisms. This implies that the convolution algebras are self-induced and the convolution modules are smooth in the sense of R. Meyer. We also show that Lie groupoid actions that are submersive, proper, and transitive have projective convolution modules. This implies that all convolution algebras are quasi-unital. We provide a long list of examples and applications, such as to bornological noncommutative tori, which are Hopf monoids in the Morita category.