TITLE: Why does sheafification functor being left adjoint imply that the presheaf kernel is a sheaf kernel
QUESTION [5 upvotes]: I want to understand Ravi Vakil's remarks to 2.4.L. I wonder why sheafification functor being left adjoint implies that the presheaf kernel is a sheaf kernel. 
By his 1.6.12, kernel, which is a limit, commutes with right adjoints. But sheafification functor is left adjoint (so I don't now why he refers to 1.6.12).

REPLY [5 votes]: The inclusion of sheaves into presheaves is a right adjoint (with left adjoint sheafification), so it preserves limits, and in particular kernels. This means that the kernel of a map of sheaves, when it exists, must also be the corresponding kernel of a map of presheaves. 
But then you need to know that kernels of sheaves exist (so that you can compute them as kernels of presheaves); for this you need to know that the inclusion of sheaves into presheaves creates limits, not just preserves them. This is true of any monadic right adjoint, which includes this case.