  Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell embedding
theorem states the existence of a ring $R$ and an exact full embedding
$\mathcal{A} \rightarrow R$-Mod. This theorem is useful as it allows one to
prove general results about abelian categories within the context of
$R$-modules. The goal of this report is to flesh out the proof of the embedding
theorem. We shall follow closely the material and approach presented in Freyd
(1964). This means we will encounter such concepts as projective generators,
injective cogenerators, the Yoneda embedding, injective envelopes, Grothendieck
categories, subcategories of mono objects and subcategories of absolutely pure
objects.
