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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

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  • $\begingroup$ 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. $\endgroup$ – Kevin Carlson Nov 14 '18 at 18:53
  • $\begingroup$ I do not follow the meaning of the middle part of your sentence: "[] just with a richer diagram shape than the discrete one" $\endgroup$ – user122424 Nov 14 '18 at 21:07
  • $\begingroup$ 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. $\endgroup$ – Kevin Carlson Nov 14 '18 at 22:05

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