# Kan extension pseudonatural transformations

Consider the 2-category $[S, H]$ of 2-functors $S\to H$ (in which, obviously, $S$ and $H$ are 2-categories). And consider a (possibly fully faithful) functor $T: S\to Z$ For simplicity, let's assume that every 2-functor $S\to H$ admits a pointwise (right) Kan extension along $T$ (for instance, we may assume that $H$ is complete). So we get a 2-functor

$Ran _T : [S, H]\to [Z,H] .$

Consider, now, the 2-category $[S, H]_{PS}$ of 2-functors and pseudonatural transformations... My questions is whether there is an extension of the 2-functor $Ran _ T$ to a 2-functor $RAN_ T : [S, H]_{PS}\to [Z,H] _ {PS}$ such that $RAN_T$ restricted to $[S,H]$ is isomorphic to $Ran _ T$.

I'm almost sure that it is possible to do so using the universal property of the Kan extension and the results given in S Lack's Codescent objects and coherence.

But I would like to know if there is a natural way of constructing such a 2-functor. And, furthermore, it would help me a lot some indications of literature about it.

After thinking (and discussing with professor Steve Lack), I am sure that this extension of the Kan extension does not exist. Such an extension would give a 2-functor $RAN_T : [S,H]_{PS}\to [Z,H]_{PS}$. And, of course, this 2-functor would take pseudonatural isomorphic diagrams to pseudonatural isomorphic diagrams. But there are pseudonatural isomorphic diagrams whose Kan extensions aren't pseudonatural isomorphic.

Therefore there is no such (strict) extension (as I originally asked for in this question).

The following example is an adaptation of an example given by professor Steve Lack.

Let $\nabla 2$ be the localization of the category $2$ (w.r.t. all morphisms). And let $\ast$ be the category with only one object and only one morphism (the identity). If S is the 2-category consisting of a parallel pair of arrows ($\alpha : a\to b$, $\beta : a\to b$) and Z is the category consisting of a parallel pair of arrows with an equalizer. $S$ and $Z$ have no nontrivial 2-cells, we can consider the following diagrams

$X: S\to Cat$ and $Y: S\to Cat$

such that $X(a)= Y(a) = \ast$, $X(b) = Y(b) = \nabla 2$, $X(\alpha ) = X (\beta )$ and $Y(\alpha ) \not = Y( \beta )$.

$X$ is pseudonaturally isomorphic to $Y$. However the Kan extensions of $X$ and $Y$ are not pseudonaturally isomorphic (because $[Z, Cat] _{PS} (Ran_T X, Ran_T Y ) = \emptyset$ ).