## $\mu$-presentable object as $\mu$-small colimit of $\lambda$-presentable objects

Remark 1.30 of Adámek and Rosický, Locally Presentable and Accessible Categories claims that in any locally $$\lambda$$-presentable category, each $$\mu$$-presentable object can be written as a $$\mu$$-small colimit of $$\lambda$$-presentable objects. I’ve also seen this stated in the literature without any reference given, suggesting it is considered “well-known to experts”.

However, as Mike Shulman pointed out in a comment on the answer https://mathoverflow.net/a/306129, it is unclear how the argument on pages 35 to 37 of Makkai and Paré cited in Remark 1.30 proves the claim. Not only is it unclear how to apply Lemma 2.5.2 of MP, but the category $$\mathbf{K}$$ constructed in its proof, which is the indexing category for the colimit produced by the lemma, has size which is not obviously bounded in terms of the sizes of the input diagrams, because it involves arbitrary morphisms between the given objects, not just ones that appear in the given diagrams.

Does anyone know how the claim of Remark 1.30 is to be proved? Alternatively, is there another, perhaps entirely different, proof in the literature?