KZ identifies the torsion quotient with Hecke modules under the Hecke-rank hypothesis

Exact localization makes ${\mathcal{O}}_{\mathrm{t}\mathrm{o}\mathrm{r}}$ Serre, and [F1]–[F2] give an exact fully faithful functor $\stackrel{}{\mathrm{KZ}}:\mathcal{C}\to \mathcal{H}\text{-mod}$ with subquotient-closed image. It is faithful on objects: a zero fiber means a zero localized bundle. Set $Q=\bar{P}$, the image of [F3]'s projective.

For every torsion object $T$, $\mathrm{Hom}(P,T)=\mathrm{KZ}(T)=0$. Consequently $P$ has no nonzero torsion quotient. In the explicit Serre-quotient roofs, any source subobject $P^{\mathrm{\prime }}\subset P$ with torsion quotient must equal $P$. A map $P\to M\mathrm{/}T$ lifts to $M$ by projectivity, uniquely modulo maps into $T$, which are zero. Hence ${\mathrm{Hom}}_{\mathcal{C}}(Q,\bar{M})={\mathrm{Hom}}_{{\mathcal{O}}_c}(P,M)=\mathrm{KZ}(M)$ naturally. This proves $Q$ projective because its represented functor is exact. It is a finite generator: evaluation from finitely many copies of $Q$ is onto, since a nonzero cokernel would receive a nonzero map from $Q$, lifting by projectivity and contradicting the evaluation image. Hom spaces are finite by [F1].

Put $A={\mathrm{End}}_{\mathcal{C}}(Q{)}^{\mathrm{o}\mathrm{p}}={\mathrm{End}}_{{\mathcal{O}}_c}(P{)}^{\mathrm{o}\mathrm{p}}$. Finite presentations by $Q$ and projectivity show that ${\mathrm{Hom}}_{\mathcal{C}}(Q,-)$ is an equivalence $\mathcal{C}\simeq A\text{-mod}$: apply it to a presentation, and conversely lift a finite free $A$-presentation to copies of $Q$ and take its cokernel. Naturality identifies the Hecke action with restriction along an algebra homomorphism $\varphi :\mathcal{H}\to A$. Explicitly if $q_h\in \mathrm{End}(P)$ acts by precomposition, $h\cdot f=f{q}_h$, then $\varphi (h)=q_h^{\mathrm{o}\mathrm{p}}$ and $\varphi (h_1{h}_2)=\varphi (h_1)\varphi (h_2)$. Under $f\mapsto f^{\mathrm{o}\mathrm{p}}$ the module $\mathrm{KZ}(P)$ is the regular left $A$-module restricted by $\varphi$.

The map $\varphi$ is surjective. Let $B=\mathrm{im}\varphi \subset A$, an $\mathcal{H}$-submodule of the regular restricted module, containing $1$. By [F1] it belongs to the essential image. Full faithfulness makes its inclusion into $A$ an $A$-linear map when interpreted through the equivalence in the preceding step. Its image is therefore an $A$-submodule containing $1$, hence all of $A$. Thus $B=A$. This proves surjectivity using actual subobject closure; exactness and the numerical hypothesis alone would not prove it.

By [F3], $\dim A=\mathrm{\mid }W\mathrm{\mid }$. The stated hypothesis gives $\dim \mathcal{H}=\mathrm{\mid }W\mathrm{\mid }$, so the surjection $\varphi$ is an isomorphism. Restriction along it is an equivalence, and the equivalence in the earlier step identifies it with $\stackrel{}{\mathrm{KZ}}$. This proves the required torsion-quotient equivalence under exactly the stated Hecke-rank hypothesis.