I having been thinking of how to formulate the idea of a universal 2-arrow (or 2-cell) in a 2-category.
As some background, I am learning about Kan extensions. In $Cat$, I understand that a left Kan extension of a functor $F:C\rightarrow D$ along a functor $K:C\rightarrow C'$ is a universal arrow $(Lan_KF,\eta:F\Rightarrow (Lan_KF)\circ K)$ from $F\in D^C$ to $K^*:D^{C'}\rightarrow D^C$. Looking up the dual notion of Kan lift on n-lab [1], the author describes "a 2-cell $p\circ \widetilde{f} \Rightarrow f$ which is universal among 2-cells of this form".
Now, I am wondering how to formulate the definition of a universal 2-cell analogous to how a universal 1-arrow from an object to a functor is defined in 1-category theory. I could not find any definition online so I tried to see if I could come up with a reasonable definition. But, I haven't been able to come up with a definition that specializes to the case of Kan extensions. Would anybody be able to give me some ideas on how to formulate such a definition?
[1] https://ncatlab.org/nlab/show/Kan+lift
*Edit: I realized it might be helpful to include the most promising definition I came up with it describe the problem with it:
Definition: A Universal 2-cell from a 1-arrow to a 2-functor: Let $F:C\rightarrow D$ be a 2-functor between 2-categories. Let $f:d\rightarrow d'\in D$. Then a universal 2-cell from $f$ to $F$ is a pair consisting of a 1-arrow $g:c\rightarrow c'\in C$ such that $Fc=d$ and $Fc'=d'$ and a 2-cell $\eta:f\Rightarrow Fg$ such that for any other 2-cell from $f$ to $F$, i.e. a pair $h:c\rightarrow c'$ and $\alpha:f\Rightarrow Fh$, there exists a unique 2-cell $\gamma:g\Rightarrow H$ of $C$ such that $\alpha=(F\gamma)\cdot\eta$.
This seems nice but there are no 2-functors involved in the definition of Kan extensions. The functor involved in the definition of Kan extensions, $K^*:D^{C'}\rightarrow D^C$ is not a 2-functor but is the image of $K:C'\rightarrow D$ under the 2-functor $D^{(-)}:Cat\rightarrow Cat$. However I can't figure out how to get the definition to work around this.
This is a wild goose chase, I’m afraid. There’s nothing inherently 2-dimensional about the universal property of a Kan extension, which is a perfectly ordinary universal object in a slice category formed from the hom categories. This is just a 2-arrow which has a universal property, not some new kind of thing called a universal 2-arrow.