A complete theory of cognitive architecture (i.e., the basic processes and modes of composition that together constitute cognitive behaviour) must explain the systematicity property--why our cognitive capacities are organized into particular groups of capacities, rather than some other, arbitrary collection. The classical account supposes: (1) syntactically compositional representations, and (2) processes that are sensitive to--compatible with--their structure. Classical compositionality, however, does not explain why these two components must be compatible, they are only compatible by the ad hoc assumption (convention) of employing the same mode of (concatenative) compositionality (e.g., prefix/postfix, where a relation symbol is always prepended/appended to the symbols for the related entities). Architectures employing mixed modes do not support systematicity. Recently, we proposed an alternative explanation without ad hoc assumptions, using category theory. Here, we extend our explanation to domains that are quasi-systematic (e.g., aspects of most languages), where the domain includes some but not all possible combinations of constituents. The central category-theoretic construct is an adjunction involving pullbacks, where the primary focus is on the relationship between processes modelled as functors, rather than the representations. A functor is a structure-preserving map (or construction, for our purposes). An adjunction guarantees that the only pairings of functors are the systematic ones. Thus, (quasi-)systematicity is a necessary consequence of a categorial cognitive architecture whose basic processes are functors that participate in adjunctions.