Consider the 2-category of locally presentable categories, cocontinuous functors, and natural transformations. I believe that this 2-category is 2-cocomplete in the sense of containing all small 2-colimits. One proof is outlined in Mike Shulman's answer to Martin Brandenburg's question [2-colimits in the category of cocomplete categories](https://mathoverflow.net/questions/64978/2-colimits-in-the-category-of-cocomplete-categories) from a couple years ago. In my paper with Alex Chirvasitu ([link](http://arxiv.org/abs/1105.3104)), we claim (with a sketch of a proof) that the following works: (i) every cocontinuous functor between presentable categories has a right adjoint, which is continuous and commutes with sufficiently-filtered colimits, and every functor of this type has a left adjoint; (ii) take the diagram whose colimit you want to compute, and consider the corresponding diagram in the category of locally presentable categories, right adjoints, and natural transformations; it suffices to compute the limit of that diagram; (iii) compute the limit of that diagram in the 2-category of all categories, and check that the limit is presentable, and that the functors involved in the limit are continuous and commute with sufficiently filtered colimits.


My question is simply a reference request: is there a paper in the published literature that provides a careful proof that the 2-category of locally presentable categories, cocontinuous functors, and natural transformations is 2-cocomplete? I'd rather cite this fact in my current project than reproduce the above argument, and I don't want to cite a paper (even my own) that only provides a "sketch of proof", if the fact is one I plan to rely on later.


The claim is not in the standard reference by [Adamek and Rosicky](http://books.google.com/books/about/Locally_Presentable_and_Accessible_Categ.html?id=iXh6rOd7of0C). The closest there is the fact that the 2-category of accessible categories and functors that commute with sufficiently filtered colimits is 2-complete (and in fact 2-limits can be computed in Cat).



Reference request: colimits of locally presentable categories
I'm a bit late to the party, but I believe there *is* a canonical reference for this fact: Greg Bird's 1984 thesis *Limits in 2-Categories of Locally Presentable Categories*. Although apparently unpublished, Google Scholar lists 20 citations to it. According to [this post](http://article.gmane.org/gmane.science.mathematics.categories/5928/match=greg+bird+thesis) by Steve Lack to the categories mailing list, Bird proves that the 2-category of locally presentable categories, left adjoint functors, and all natural transformations has all [flexible 2-limits](https://ncatlab.org/nlab/show/flexible+limit) and all flexible 2-colimits. This includes all bi(co)limits.


I don't know the history, but I gather that Bird's thesis work was in fact closely related to the very development of the notion of a flexible 2-limit: these are the strict 2-limits whose weights are such that they don't don't actually demand anything too strict, the "homotopically meaningful" strict 2-limits. Lack [later showed](https://eudml.org/doc/232481) that in fact they are the cofibrant objects in an appropriate 2-model structure on the 2-category of weights. 


Unfortunately, I haven't been able to find a copy of Bird's thesis online. **Update:** Ross Street has now made it available again [(link)](http://maths.mq.edu.au/~street/BirdPhD.pdf); see comment below. 


