This is related to my other question, but I think now I have enough of an understanding of the topic to better phrase it.
In Stephen Lack's paper "A 2-Categories Companion", he mentions that there are two viewpoints of 2-categories (and, more generally I suppose, bicategories): 2-categories as a generalized category, and 2-categories as a collection of "category-like things". For this question, I am interested in the latter approach.
Lack goes on in section 2.2 of his paper to note how, in terms of this second view of 2-categories, the notion of a (co)limit can be defined for an object in a 2-category.
So, given this notion's natural extension to bicategory theory, my question is, do we have the usual theorem that lax functors between bicategories with a right adjoint preserve limits in this sense, and dually that lax functors between bicategories preserve colimits?
I assume that a result like this is most likely known in the folklore at least, but I'm trying to find a reference amongst the literature on bicategories, and have yet to find anything explicit about this.