Let $F,G, H: Mod \to Mod$ be three left exact functors such that $R^iF(-)\cong R^iG(-)$ for all $i\in\mathbb{N}$. We consider the exact sequence $\cdots\to R^iF(M)\to R^iG(M)\to R^iH(M)\to R^{i+1}F(M)\to R^{i+1}G(M)\to R^{i+1}H(M)\to\cdots$ where $M$ is an $R$-module ($R$ is a commutative Noetherian ring).
From the above exact sequence, can we have $R^iH(M)=0$?