I am currently reading Robert Friedman's notes on sheaf cohomology and hypercohomology http://www.math.columbia.edu/~rf/cohomology.pdf
In it he defines quasi-isomorphism of complexes of sheaves by saying the $i^{th}$ cohomology sheaves $\mathcal{H}^i$ are isomorphic.
I can't seem to find a precise definition of these. Does he mean that the cohomology of the $i^{th}$ sheaf in one complex is the same as the cohomology of the $i^{th}$ sheaf in the other complex, for all $i$.
I would be grateful if someone could provide a reference?
The reason i am asking is motivated by the case when the complexes are unbounded (which if i'm not mistaken is not covered in these notes).
Splatenstein in http://www.numdam.org/article/CM_1988__65_2_121_0.pdf introduces $K$-injective resolutions of complexes of objects in an abelian category (not necessarily bounded) and defines cohomology for them as the cohomology of the complex of global sections of a $K$ injective resolution.
At the start of the paper he says quasi-isomorphism of two complexes $A^*$, $B^*$ is a map between the complexes inducing isomorphism between $H^*(A^*)$ and $H^*(B^*)$. How does this look like in the case when $A^*$ and $B^*$ are complexes of sheaves?