TITLE: Is the category of left exact functors abelian?
QUESTION [10 upvotes]: Let us consider the category of left exact functors between two abelian categories. Is this category abelian?
My intuition is that it is not. Does someone have any counterexample ? Or any proof that I'm false?
I think that we will have problems with cokernels...

REPLY [3 votes]: This is really a comment rather than an answer.  I just wanted to give a link to my answer on MathOverflow https://mathoverflow.net/questions/299014/is-the-category-of-left-exact-functors-abelian , which contains an example showing that the category of left exact functors $\mathcal A^{op}\to (k{-}\mathrm{vect})^{op}$ does not need to be abelian, when $\mathcal A$ is a small (really, small) abelian category and $k{-}\mathrm{vect}$ is the category of vector spaces over a field $k$.