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.


Thank you in advance!



Kan extension pseudonatural transformations
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 $[Z, Cat] \_{PS} (Ran\_T X, Ran\_T Y ) = \emptyset$ ).


