Let $\mathcal{C}$ be the category of modules over a ring. 
Let also $\mathcal{F}$ be a class of objects in $\mathcal C$ closed under pure subobject (pure quotient) and direct limit. Is $\mathcal{F}$ closed under pure quotient (pure subobject)? 

What can be said about the category of quasi-coherent sheaves?