Tile systems offer a general paradigm for modular descriptions of concurrent systems, based
on a set of rewriting rules with side-effects. Monoidal double categories are a natural
semantic framework for tile systems, because the mathematical structures describing system
states and synchronizing actions (called configurations and observations, respectively, in our
terminology) are monoidal categories having the same objects (the interfaces of the system).
In particular, configurations and observations based on net-process-like and term structures
are usually described in terms of symmetric monoidal and cartesian categories, where the
auxiliary structures for the rearrangement of interfaces correspond to suitable natural
transformations. In this paper we discuss the lifting of these auxiliary structures to double
categories. We notice that the internal construction of double categories produces a
pathological asymmetric notion of natural transformation, which is fully exploited in one
dimension only (for example, for configurations or for observations, but not for both).
Following Ehresmann (1963), we overcome this biased definition, introducing the notion of
generalized natural transformation between four double functors (rather than two). As a
consequence, the concepts of symmetric monoidal and cartesian (with consistently chosen
products) double categories arise in a natural way from the corresponding ordinary versions,
giving a very good relationship between the auxiliary structures of configurations and
observations. Moreover, the Kelly–Mac Lane coherence axioms can be lifted to our setting
without effort, thanks to the characterization of two suitable diagonal categories that are
always present in a double category. Then, symmetric monoidal and cartesian double
categories are shown to offer an adequate semantic setting for process and term tile systems.