# Nearly locally presentable categories

Here1, in the remark $$2.3 (1)$$ how from the fact that $${\cal K}(A,-)$$ does not preserve coproducts it follows that $${\cal K}(A,-)$$ sends special $$\lambda$$-directed colimits to $$\lambda$$-directed colimits and not to special $$\lambda$$-directed ones?

1 Leonid Positselski, Jiří Rosický: Nearly locally presentable categories, Theory and Appl. of Categories 33 (2018), #10, p.253-264; http://www.tac.mta.ca/tac/volumes/33/10/33-10abs.html https://arxiv.org/abs/1710.10476

• To preserve special directed colimits, which are by definition always coproducts, just with a richer diagram shape than the discrete one, you would have to preserve coproducts. – Kevin Carlson Nov 14 '18 at 18:53
• I do not follow the meaning of the middle part of your sentence: "[] just with a richer diagram shape than the discrete one" – user122424 Nov 14 '18 at 21:07
• It's a coproduct expressed as a filtered colimit of sub-coproducts of up to size $\lambda$ rather than as a discrete colimit. But it's still required to be a coproduct. – Kevin Carlson Nov 14 '18 at 22:05