  Let $\mathcal{A}$ be an essentially small abelian category. We prove that if
$\mathcal{A}$ admits a generator $M$ with ${\rm End}_{\mathcal{A}}(M)$ right
artinian, then $\mathcal{A}$ admits a projective generator. If $\mathcal{A}$ is
further assumed to be Grothendieck, then this implies that $\mathcal{A}$ is
equivalent to a module category. When $\mathcal{A}$ is Hom-finite over a field
$k$, the existence of a generator is the same as the existence of a projective
generator, and in case there is such a generator, $\mathcal{A}$ has to be
equivalent to the category of finite dimensional right modules over a finite
dimensional $k$-algebra. We also show that when $\mathcal{A}$ is a length
category, then there is a one-to-one correspondence between exact abelian
extension closed subcategories of $\mathcal{A}$ and collections of
Hom-orthogonal Schur objects in $\mathcal{A}$.
