# Why are the left exact functors from an abelian category to abelian groups cocomplete and have a injective generator?

Let $\mathcal{C}$ be an abelian category, $\mathcal{Ab}$ the category of abelian groups and $Lex(\mathcal{C}, \mathcal{B})$ the category of left exact functors between abelian categories.

What is the simplest way (or at least a way) to prove that $Lex(\mathcal{A},\mathcal{Ab})$ is cocomplete and has an injective Cogenerator?

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OK, $\mathcal C$ is an abelian category, but what's this relevant for? I think this question is very poorly stated, and I also think that a proof of this fact can be quickly found in the standard literature on the topic. – Fernando Muro May 23 '13 at 9:45
This was a typo, i am sorry. It is corrected now. – Harald May 25 '13 at 10:41