The classical account for systematicity of human cognition supposes: (1) syntactically compositional representations; and (2) processes that are sensitive to their structure. The problem with this account is that there is no explanation as to why these two components must be compatible, other than by ad hoc assumption (convention) to exclude nonsystematic variants that, e.g., mix prefix and postfix concatenative compositional schemes. Recently, we proposed an alternative explanation (Phillips & Wilson, 2010) without ad hoc assumptions, using a branch of mathematics, called category theory. Here, we extend our explanation to domains that are quasi-systematic (e.g., language), where the domain includes some but not all possible combinations of constituents. The central category-theoretic construct is an adjunction involving pullbacks, where the focus is on the relations between processes, rather than the representations. In so far as cognition is systematic, the basic building blocks of cognitive architecture are adjunctions by our theory.