I'm following Vakil's notes - chapter on category theory. One issue that is unclear in the notes is the conclusion that right adjoint functors are left exact. The notes define a left exact functor as a functor that preserves left exactness. A sequence $A\xrightarrow{f}B\xrightarrow{g}C $ is exact at $A$ if $\text{im } f = \ker{g}$ in the categorical sense: We cannot use argumens involving elements of sets.
The notes show that right adjoints commute with limits. Since kernels are limits, it is sufficient to show that commuting with kernels implies left exactness.
Using the definitions above, how to show that an additive functor that commutes with kernels is left exact?
I've gone through the first 3 chapters of Maclane's Categories. This is my background. I'm interested in a general argument in abelian categories. I understand that this can be reduced to an argument in modules, but I want to understand how this works in general.
Thanks