  Categorical structures and their pseudomaps rarely form locally presentable
2-categories in the sense of Cat-enriched category theory. However, we show
that if the categorical structure in question is sufficiently weak (such as the
structure of monoidal, but not strict monoidal, categories) then the 2-category
in question is accessible. Furthermore, we explore the flexible limits that
such 2-categories possess and their interaction with filtered colimits.
