Let $f:\mathbb{C} \to \mathbb{D}$ be a functor of 2-categories and let $\operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$ denote the 2-category of functors and lax natural transformations.


Is it true in general that restriction along $f$,


$f^\*:\operatorname{Fun}(\mathbb{D},\operatorname{Cat})^{\operatorname{lax}} \to \operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$


preserves all lax (weighted) limits?



Preservation of lax limits in categories of functors and lax natural transformations
I believe this functor preserves all *oplax* limits, and more generally all "$l$-rigged limits" in the sense of [Enhanced 2-categories and limits for lax morphisms](https://arxiv.org/abs/1104.2111) by Steve Lack and myself. (I don't think it's reasonable to ask it to preserve *lax* limits, since these 2-categories don't in general even *have* lax limits.)


To prove this, using the technology of that paper, observe that $\mathrm{Fun}({C},\mathrm{Cat})^{\rm lax}$ is $T\_{{C}}\rm Alg\_l$ for a 2-monad $T\_{{C}}$ on the 2-category $\mathrm{Cat}^{\mathrm{ob}{C}}$ of families of categories indexed by the objects of ${C}$. This 2-category underlies an $\mathcal{F}$-category $T\_C \mathbb{A}\mathrm{lg}\_l$ for a corresponding $\mathcal{F}$-monad $T\_{C}$ on $\mathrm{Cat}^{\mathrm{ob}{C}}$, which includes the data of the strict natural transformations as well and their embedding into the lax ones.


The main result of the above-cited paper is that the forgetful $\mathcal{F}$-functor $U\_C:T\_C \mathbb{A}\mathrm{lg}\_l\to\mathrm{Cat}^{\mathrm{ob}{C}}$ creates $l$-rigged limits, so that $T\_C \mathbb{A}\mathrm{lg}\_l$ has these limits and $U\_C$ preserves them. These are a class of $\mathcal{F}$-enriched weighted limits which include "oplax limits whose generating projections are strict and detect strictness". (Steve had already proven in [Limits for Lax Morphisms](https://link.springer.com/article/10.1007/s10485-005-2958-5) that oplax limits of this sort lift in the 2-categorical case.)


Now $f:C\to D$ induces a commutative square of $\mathcal{F}$-functors involving $U\_C$ and $U\_D$ and the restriction functors $f^\*: \mathrm{Cat}^{\mathrm{ob}{D}} \to \mathrm{Cat}^{\mathrm{ob}{C}}$ and $f^\* :T\_D \mathbb{A}\mathrm{lg}\_l \to T\_C \mathbb{A}\mathrm{lg}\_l$. The former $f^\*$ preserves all limits since they are pointwise, hence the composite $T\_D \mathbb{A}\mathrm{lg}\_l \to \mathrm{Cat}^{\mathrm{ob}{D}} \to \mathrm{Cat}^{\mathrm{ob}{C}}$ preserves $l$-rigged limits. Therefore, so does the composite $T\_D \mathbb{A}\mathrm{lg}\_l \to T\_C \mathbb{A}\mathrm{lg}\_l\to \mathrm{Cat}^{\mathrm{ob}{C}}$. This means that the comparison maps for $l$-rigged limits induced by $f^\* :T\_D \mathbb{A}\mathrm{lg}\_l \to T\_C \mathbb{A}\mathrm{lg}\_l$ are inverted by $U\_D$. But the forgetful $\mathcal{F}$-functor $U\_D$ is conservative, so these comparison maps are already isomorphisms in $T\_D \mathbb{A}\mathrm{lg}\_l$, i.e. $f^\*$ preserves $l$-rigged limits.


