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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 $Lex(\mathcal{C},\mathcal{Ab})$ is cocomplete and has an injective cogenerator?

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  • $\begingroup$ 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. $\endgroup$
    – Fernando Muro
    Commented May 23, 2013 at 9:45
  • $\begingroup$ This was a typo, i am sorry. It is corrected now. $\endgroup$
    – Harald
    Commented May 25, 2013 at 10:41

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A complete proof, albeit in German, is given in Section 4.13 of B. Pareigis, Kategorien und Funktoren, Teubner, Stuttgart (1969).

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