  The present note has three aims. First, to complement the theory of cofibrant
generation of algebraic weak factorisation systems (AWFSs) to cover some
important examples that are not locally presentable categories. Secondly, to
prove that cofibrantly KZ-generated AWFSs (a notion we define) are always lax
orthogonal. Thirdly, to show that the two known methods of building lax
orthogonal AWFSs, namely cofibrantly KZ-generation and the method of "simple
adjunctions", construct different AWFSs. We study in some detail the example of
cofibrant KZ-generation that yields representable multicategories, and a
counterexample to cofibrant generation provided by continuous lattices.
