# Is there an analogue of final functors for genuine 2-categorical limits

A functor $$I\to J$$ of $$1$$-categories is called final, if each undercategory $$(j, I)$$ is connected.

More generally, for $$(\infty,1)$$-categories there is an analogous notion where one requires the undercategories $$I_{j/}$$ to be weakly contractible.

In both of these cases, the main property is that final functors $$I\to J$$ identify $$J$$-colimits with $$I$$-colimits. This also handles the case of pseudo colimits in $$2$$-categories since these don't make use of non-invertible $$2$$-cells (see this question).

My question is about the genuine 2-categorical case:

Given a diagram (i.e. a $$2$$-functor) $$F\colon J\to \mathcal C$$ in some $$2$$-category, where $$J$$ is equipped with a set of marked $$1$$-cells. A $$\sigma$$-limit (see, e.g., this paper) of $$F$$ is a universal co-cone which commutes up to specified $$2$$-cells and such that each such $$2$$-cell which lives over a marked arrow is invertible. (This specializes to lax limits by marking only identities and to pseudo limits by marking all arrows in $$J$$).

Is there a good notion of a final $$2$$-functor $$f\colon I\to J$$ between $$2$$-categories such that $$f$$ will identify $$J$$-shaped $$\sigma$$-limits with $$I$$-shaped $$\sigma$$-limits?

EDIT: In the very paper I linked, final diagrams $$I\to J$$ are treated in the case where $$I$$ is "$$\sigma$$-filtered" in an ad-hoc way and not using any kind of undercategories. I would be interested in a more conceptual definition which makes sense for general $$2$$-functors.