
> 
> Let $\mathcal{A} \subset \mathcal{K}$ be two locally presentable
>  categories. $\mathcal{A}$ reflective and closed under filtered
>  colimits. Then $\mathcal{A}$ is a small-orthogonality class. Let
>  $R:\mathcal{K}\to \mathcal{A}$ be the reflection. Let $G$ be a dense
>  generator of $\mathcal{K}$ consisting of objects not belonging to
>  $\mathcal{A}$. Can we conclude that $\mathcal{A}$ is the
>  small-orthogonality class with respect to the set of maps 
>  $\{\eta\_g:g\to Rg \mid g\in G\}$ ?
> 
> 
> 


I think that the answer is negative in full generality and I would like to see a counterexample.


Note: I have edited my question to remove the case $G\subset \mathcal{A}$.



About small-orthogonality classes of a locally presentable category
It seems to me that a negative answer follows immediately from the possibility, in many cases, of choosing $G\subset \mathcal A$, so that $G$ doesn't know anything about $R$ until one closes it under some colimits. Given such a $G$, every object of $\mathcal K$ is right orthogonal to each $\eta\_g$, since these are isomorphisms.


For instance, let $\mathcal A$ be torsion-free abelian groups, which are reflective and closed under filtered colimits in abelian groups, $\mathcal K$. Now $\mathcal K$ has a dense generator consisting of torsion-free groups; for instance, the singleton $\mathbb{Z}^2$ suffices.


