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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?

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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 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 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.

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    $\begingroup$ Maybe worth noting that this argument depends crucially on the use of $\mathcal{F}$-categories (or something equivalent), since the forgetful 2-functor $T_C \mathrm{Alg}_l \to \mathrm{Cat}^{\mathrm{ob}C}$ is not conservative: a lax natural transformation can have invertible components without being itself invertible. We need to know that the comparison maps are in fact strict natural transformations, and this requires knowing that the oplax limits are in fact $\mathcal{F}$-limits, i.e. their generating projections are strict and detect strictness. $\endgroup$
    – Mike Shulman
    Commented Jul 19, 2023 at 16:27

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