Is the the shift functor exact

Question

Given a cochain complex $C^{\bullet}$, are the notations $H^p(C^{\bullet}[n])$ and $H^p(C^{\bullet})[n]$ equivalent or do they differ? We know that $H^p(C^{\bullet}[n])=H^{p+n}(C^{\bullet})$ but $H^p(C^{\bullet})[n]$ seems to have two different interpretations. The first interpretation is as the cohomology group $H^{p}(C^{\bullet})$ concentrated in degree $-n$ and the second interpretation is $H^p(C^{\bullet})[n]=H^{n+p}(C^{\bullet})$. Or are both interpretations equivalent even though I do not see them as equivalent? Can someone elaborate on the different points of view.