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!


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 $[Z, Cat] _{PS} (Ran_T X, Ran_T Y ) = \emptyset$ ).


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