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.


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