sentences:  Morphisms in the derived category are obtained from the homotopy category of chain complexes of sheaves by taking the zeroth cohomology of the complex, i.e.For the other side of the Verdier duality statement above, we have to take for granted the fact that when "X" is a compact orientable "n"-dimensional manifold 
  which is the dualizing complex for a manifold.Now we can re-express the right hand side as
  We finally have obtained the statement that
  By repeating this argument with the sheaf "k"X replaced with the same sheaf placed in degree "i" we get the classical Poincaré duality