Dear Redi Haderi Max Kelly's book ``Basic concepts of enriched category theory'' generalises this whole Kan business to V-categories for a decent base V. So, for the case V = Cat, you obtain the result for 2-categories. I say ``the result'', but of course, for 2-categories there are versions for weaker structures (bicategories, pseudofunctors, and so on). The Cat-enriched version is the strict case which you seem to want. Regards, Ross Street On 16 Feb 2020, at 6:11 AM, Redi Haderi <haderiredi@gmail.com<mailto:haderiredi@gmail.com>> wrote: Given a category C we know that the category of set-valued presheaves on C may be interpreted as a free cocompletion of C. More precisely, if we denote PSh(C) the presheaf category, then colimit-preserving functors from PSh(C) to a cocomplete category D correspond to functors from C to D (via Yoneda extension). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]