**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.