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Alastair Grant-Stuart (Jun 24 2020 at 19:24):Say I'm given a 2-category whose objects are categories equipped with some extra structure, and whose 1-cells (2-cells) are functors (resp. natural transformations) that respect the extra structure. So by analogy to 1-categories and sets, I might say that is a "concrete" 2-category.
I'd like to know about pointwise extensions in . In particular, my question is:
Are there analogues in of the (co)limit or (co)end formulae for Kan extensions in ?
I'm aware of this paper by Street & Walters on Yoneda structures, and that this can be used to define what 'pointwise' means here. I've also seen mention of some other approaches to pointwise extensions on nLab, involving either proarrows or comma objects.
But I'm still new to CT, and don't yet know enough to tell which of these directions to learn about first. So, I thought I should check with the experts: which of these approaches, or any others you may know of, is the best way to approach my question?
(For my practical purposes, we can take "best" to mean easiest for a beginner, rather than most general or elegant....)