On the lack of colimits in various categories arising in pointfree topology and algebraic logic

We prove that the category of McKinsey-Tarski algebras is not equivalent to a variety of algebras, thus answering a question of Peter Jipsen in the negative. More generally, we show that various categories of BAOs (boolean algebras with an operator), Heyting algebras, and frames with appropriate morphisms between them are not cocomplete. As a consequence, none of these categories is equivalent to a prevariety, let alone a variety.