Dear Keith Feeling sure that the quasicategory people would have such a result, I asked Alexander Campbell who kindly replied: ``I don't remember a reference for bicategories, but references for quasi-categories are easier to locate: Theorem 5.1.5.6 (on page 345) of Lurie's book 'Higher topos theory' and Theorem 6.3.13 (on page 288) of Cisinski's book 'Higher categories and homotopical algebra' both state that the functor (y_A)^* :Cocts(Psh(A),C) --> Fun(A,C) is an equivalence for any small simplicial set A and cocomplete quasi-category C. '' Now back to me: As far as bicategories are concerned, the fact that the 2-category Hom(C^{op},Cat) of pseudofunctors C^{op}\to Cat is the free cocompletion up to biequivalence of the small bicategory C is a fairly routine generalization of the category case. The basic ingredient is the notion of (bicategorical) colimit colim(J,F) of a pseudofunctor F : C \to K weighted by a pseudofunctor J : C^{op}\to Cat; see [14. Fibrations in bicategories, Cahiers de topologie et g\'eom\'etrie diff\'erentielle 21 (1980) 111--160] The extension of F to a weighted colimit preserving pseudofunctor is colim(-,F) : Hom(C^{op},Cat) \to K when K has small weighted colimits. Also of interest, a few years ago Ren\'e Guitart pointed out a lax cocompletion property of a familiar construction (not quite presheaves or prestacks) in his talk ``Sur le foncteur diagramme'' <http://archive.numdam.org/article/CTGDC_1973__14_2_153_0.pdf>. Ross On 20 Feb 2020, at 7:16 AM, Keith Harbaugh <keith.harbaugh@gmail.com<mailto:keith.harbaugh@gmail.com>> wrote: Dear Ross, can you recommend a good source for the extension of this result to the setting of weaker notions of categories and functors? On Wed, Feb 19, 2020, 13:35 Ross Street <ross.street@mq.edu.au<mailto:ross.street@mq.edu.au>> wrote: 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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]