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!