I know this question is absolutely trivial, but having self-studied the subject I feel extremely unsure on the basics. Do not hesitate to downgrade the question, if you feel it deserves so.

Given a morphism of complex of sheaves $\varphi:\mathcal{F}^\bullet\to \mathcal{G}^\bullet$ on a topological space $X$ (eventually as nice as needed), we have two induced morphisms, one in local cohomology, $\mathcal{H}^\bullet(\varphi):\mathcal{H}^\bullet(\mathcal{F}^\bullet)\to \mathcal{H}^\bullet(\mathcal{G}^\bullet)$, and one in global cohomology $H^\bullet(\varphi):H^\bullet(X;\mathcal{F}^\bullet)\to H^\bullet(X,\mathcal{G}^\bullet)$.

The question is: does $\mathcal{H}^\bullet(\varphi)=0$ imply $H^\bullet(\varphi)=0$?

I guess so, since there is a spectral sequence with $E_2$-page given by cohomology with cefficients in local cohomology abutting to global cohomology. But as I said, I do not trust myself too much on self-study..