this question concerns the canonical bundle $\omega_X$ on a smooth projective $k-$variety $X$.
I want to consider the diagonal embedding
$i:X\rightarrow X\times X$
Furthermore denote by $p$ the projection $X\times X \rightarrow X$ on the first factor.
Now my question:
does it hold that
$i_*\omega_X \simeq p^* \omega_X$
canonically as sheaves on $X\times X$?
I thank you very much!