The regular category embedding theorem

**Abstract:**

I have given two apparently different proofs of the regular category embedding
theorem. The first, gotten by adapting Lubkin's argument for
abelian categories, is rather opaque. The second, gotten by adapting
Mitchell's proof is much more elegant. Mitchell used Grothendieck's
theorem that an AB5 category with a generator has an injective
cogenerator. However, the analogous result for regular categories
fails. It turns out that full injectivity is not needed.
Surprisingly, it turns out that ``under the hood'' the two proofs are
really doing much the same thing. It is using functors rather than
representing diagrams that makes the difference.