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Let $\mathcal{A}$ be an Abelian category. $\mathcal{L}$ is the category of absolutely pure objects of $\mathcal{A}$ and $\mathcal{L}(\mathcal{A})$ is the category of the exact left functors of $\mathcal{A}$ into $\mathcal{Ab}$ (category of abelian groups). The Theorem 7.31 of the book Freyd (Abelian Categories and Introduction to the theory of functors) says:

  • $\mathcal{L}$ is abelian and every object has an injective envelope

Freyd says (on page 149, before theorem 7.32) that by this proposition it can be concluded that he $\mathcal{L}(\mathcal{A})$ has a injective envelope. But I do not understand why this can be concluded from this statement.

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    $\begingroup$ You have also asked the same question on MathOverflow. To avoid getting the same answers on both sites, you should decide whether MathSE or MathOverflow is the right site for your question, and self-delete your question on the other site. $\endgroup$ – Geoffrey Trang Aug 19 '19 at 4:09
  • $\begingroup$ And if you feel that MO is a better place for this question, comment on Jeremy Rickard's answer to get it re-posted there. $\endgroup$ – David Roberts Aug 19 '19 at 9:54
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You've misunderstood what Theorem 7.31 applies to. It's not about absolutely pure objects of $\mathcal{A}$, the small abelian category that will be embedded in a module category. In fact, "absolutely pure" only makes sense relative to a subcategory $\mathcal{M}$.

Theorem 7.31 considers an AB5 abelian category $\mathcal{B}$ (Freyd uses the term "Grothendieck category" in a way that would now be nonstandard) such that every object embeds in an injective object, and a full subcategory $\mathcal{M}$ satisfying certain conditions, and proves that the full subcategory $\mathcal{L}$ of absolutely pure objects of $\mathcal{M}$ is abelian with injective envelopes. This can't apply directly to $\mathcal{A}$, as a small category can't be AB5.

Then, before Theorem 7.32, he applies this with $\mathcal{B}$ the category of additive functors from $\mathcal{A}$ to abelian groups, and $\mathcal{M}$ the category of functors that preserve monomorphisms. He has previously shown that in this case the conditions of Theorem 7.31 are satisfied, and that the absolutely pure objects are the left exact functors.

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