I don't have so much experience in category theory so my question may be stupid and non-sense.

1. There is a classical adjunction in algebraic geometry between the $M\rightarrow M^{\sim}$ and the global section in the affine case . 

2. We know that if we deal with quasi-coherent sheaves is an equivalences of categories 

can we prove 1 by the following way :

a) by the adjoint functor theorem $M\rightarrow M^{\sim}$ have an adjoint

b) since in a subcategory we know his adjoint (the global section) we can generalize this to the all category ( maybe by proving that there is an unique way to extend the global section functor...) 

I know that is clearly not the simple way to do that but I want to improve my category theory skill.

thanks in advance !