# Do left adjoint lax functors preserve colimits?

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.

• The question and the approach don't seem to match up. It seems more natural to me to ask where a left adjoint between bicategories preserves colimits in the bicategory, not colimits in an object in the bicategory. Am I misunderstanding you? – Kevin Carlson Sep 28 '17 at 22:29
• No, you're understanding me correctly. The reason I'm considering the latter instead of the former is that I have a particular 2-category (call it C) in mind, whose objects I'm thinking of as "category-like objects", and I have a functor F : Cat -> C which does not preserve colimits, but appears to send regular colimits to bi-colimits in C, and I'm wondering if there's any hope of viewing this F as a free functor of some sort (since it is clearly not a 1-categorical functor, if we view C as a 1-category). – Nathan BeDell Sep 29 '17 at 1:14
• Howver, I think I might need to consider lax functors between 2-categories in general, so I've updated my question to reflect this. – Nathan BeDell Sep 29 '17 at 1:15