**Question 1:** Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind\_\kappa(F): Ind\_\kappa(C) \to Ind\_\kappa(D)$ also conservative? 


**Terminology:** I think this is pretty self-explanatory, but to be clear:


* $\kappa$ is a regular cardinal. Things are perhaps most familiar when $\kappa = \aleph\_0$.
* A $\kappa$-cocomplete category is a category with $\kappa$-small colimits, i.e. colimits indexed by categories with fewer than $\kappa$-many morphisms.
* A $\kappa$-cocontinuous functor is a functor preserving $\kappa$-small colimits.
* $Ind\_\kappa(C)$ is obtained from $C$ be freely adjoining $\kappa$-filtered colimits, or by the formula $Ind\_\kappa(C) = Fun\_\kappa(C^{op},Set)$, where $Fun\_\kappa(A,B)$ is the category of $\kappa$-continuous functors from $A$ to $B$.
* The induced functor $Ind\_\kappa(F)$ is defined by left Kan extension along the Yoneda embedding.


I'm a little worried that Question 1 is too much to ask for, so here's an even milder variant.


* Let $\kappa$ be inaccessible.
* Let $Pr^L(\kappa)$ be the category (really a $(2,1)$-category) of categories which are locally presentable with respect to the universe $V\_\kappa$. That is, $Pr^L(\kappa)$ consists of categories of the form $Ind\_\lambda^\kappa(C)$ where $C$ is a $\kappa$-small and $\lambda$-cocomplete category with $\lambda < \kappa$; here $Ind\_\lambda^\kappa$ is the free cocompletion under $\kappa$-small, $\lambda$-filtered colimits. The morphisms are left adjoint functors.
* Similarly, let $Pr^L$ be the $(2,1)$-category of locally presentable categories and left adjoint functors.
* Then we have a functor $Ind\_\kappa: Pr^L(\kappa) \to Pr^L$.


**Question 2:** Does the functor $Ind\_\kappa: Pr^L(\kappa) \to Pr^L$ preserve conservative functors?


Question 2 is asking whether the property of a left adjoint between locally presentable categories being conservative depends on which universe we work in.


In the setting of either question, it's clear that if $Ind\_\kappa(F)$ is conservative, then $F$ is conservative. So in either case, if the answer to the question is affirmative, then we have "$F$ conservative $\Leftrightarrow$ $Ind\_\kappa(F)$ conservative", which would be reassuring.



Can conservativity depend on the universe?
Probably not the best one can do, and what follows might be a bit 'overkill', but it answer the question about dependency on universe, and it is a nice argument. 


Also if you know how the proof of the left transfer results I will use below works, it might give some idea on how to prove more general case of the result.


**Theorem:** Let $\kappa$ be an uncountable regular cardinal. Let $F: \mathcal{C} \to \mathcal{D}$ be a strongly $\kappa$-accessible left adjoint functor between locally $\kappa$-presentable categories. Then $F$ is conservative if and only if the restriction of $F$ to the full subcategory of $\kappa$-presentable objects is conservative.


(Here strongly $\kappa$-accessible = sends $\kappa$-presentable objects of $\mathcal{C}$ to $\kappa$-presentable objects of $\mathcal{D}$)


**Proof:** The weak factorization system (isomorphisms, all maps) on $\mathcal{D}$ is ($\kappa$-)combinatorial: it is generated by the empty set of maps. 
Hence the left transfered weak factorization system on $\mathcal{C}$ along $F$ exists and is also $\kappa$-combinatorial (I mean by this that is is cofibrantly generated by maps between $\kappa$-presentable objects).
By definition, the left class of this weak factorization system is the set of maps such that $F(i)$ is an isomorphism. But if the restriction of $F$ to $\kappa$-presentable objects is conservative it means that all generators are isomorphisms hence, the left class only contains isomorphisms, so $F$ is conservative.


**Note:** the finer version of the left transfer theorem that I'm using which specify the presentability rank can be found as Theorem B.8.(4) in [this paper](https://arxiv.org/abs/2005.02360) of mine, but at the end of day it mostly follows from an analysis of the proof of the existence of left transfer available in the litterature (for e.g. in Makkai and Rosicky [Cellular categories](https://arxiv.org/abs/1304.7572).)


