  Let $F$ be a non-archimedean local field and $G$ be a connected reductive
group over $F$. For a Bernstein block in the category of smooth complex
representations of $G(F)$, we have two kinds of progenerators: the compactly
induced representation $\text{ind}_{K}^{G(F)} (\rho)$ of a type $(K, \rho)$,
and the parabolically induced representation $I_{P}^{G}(\Pi^{M})$ of a
progenerator $\Pi^{M}$ of a Bernstein block for a Levi subgroup $M$ of $G$. In
this paper, we construct an explicit isomorphism of these two progenerators.
Moreover, we compare the description of the endomorphism algebra
$\text{End}_{G(F)}\left(\text{ind}_{K}^{G(F)} (\rho)\right)$ for a depth-zero
type $(K, \rho)$ by Morris with the description of the endomorphism algebra
$\text{End}_{G(F)}\left(I_{P}^{G}(\Pi^{M})\right)$ by Solleveld, that are
described in terms of affine Hecke algebras.
