Let $(R,m)$ a Noetherian local ring of dimension $d$. Suppose $F^\bullet, G^\bullet \colon \operatorname{Mod} \to \operatorname{Mod}$ are two cohomological $\delta$-endofunctors on the category of $R$-modules. When restricted the subcategory of modules of finite length those functors are naturally isomorphic.
Is it true that $G^n(M) \cong F^n(M)$ for any finitely generated $R$-module $M$ and $n \geq 0$?
I'm interested in the statement because the local duality $H^i_m(M) \cong \operatorname{Hom}(\operatorname{Ext}^{d-i}(M, \omega_R), E(k))$ easily follows from it.
Perhaps, something of this sort is known in the literature. That incudes variations of the statement with additional assumptions on $F$ and $G$, or versions for non-commutative rings (perhaps, even general version for an abelian category). Any reference is appreciated.