Suppose that $\mathcal{C}$ is an abelian $k$-linear category ($k$ a field) in which every object is of finite length and every $k$-vector space $\text{Hom}(X,Y)$ is finite dimensional.
How does one prove the fact that for every object $X$ of $\mathcal{C}$, the full subcategory $X^*$ of $\mathcal{C}$ whose objects are subquotients of the $X^n$ has a projective generator?