# How to use adjointness properties of sheafification to show limits/colimits are preserved

This is probably an extremely elementary question, and I am not sure if it is down to my lack of understanding of sheafification, or my lack of understanding of what it means for a right (respectively left) adjoint to preserve limits (respectively colimits).

Suppose I have sheaves of abelian groups $\mathcal{F}$ and $\mathcal{G}$. Say I want to show that the presheaf defined on open sets $U$ by $U \mapsto \mathcal{F}(U) \oplus \mathcal{G}(U)$ is itself a sheaf. I would like to do this using adjointness properties of sheafification and the forgetful functor. However, it seems to work the opposite to what I would expect. Is it correct that all I need to do is sheafify this presheaf, show that the resulting sheaf satisfies the universal property for direct sum (hence is a limit) in the category of sheaves, then apply the forgetful functor and claim that the result must be the original prehseaf?

My problem is that I know in general that limits play nice with sheafificaioin, not colimits. So say I had a family $\{ \mathcal{F}_{i} \}_{i \in I}$ of sheaves. Now suppose I defined a presheaf via the colimit $$U \mapsto \lim_{\rightarrow} \mathcal{F}_{i}(U)$$ Since sheafification is a left adjoint, it preserves colimits. So why isn't the sheafificaton of this presheaf colimit simply the sheaf colimit, which would imply it is a sheaf - a 'fact" I know to be false. I seem to be understanding "preserves (co)limit" as the inverse of what it actually means in some sense. Why do we use the forgetful functor properties and not sheafification?

I apologize if this is a poorly stated question, but I have some vague understanding, but I feel like I'm just not seeing it properly. Any explanation of these examples, or really any general comments would be appreciated.

• You should absolutely go through the exercise of proving that right/left adjoints preserve limits/colimits. You should prove it for various formulations of adjunction and (co)limits. This is a fundamental fact with which you should become intimately familiar. Aug 7, 2017 at 2:56