I'm reading "The Geometry of Moduli Space of Sheaves" (Huybrechts,Lehn). He gives a new definition of the dual sheaf:
$\ $Let E be a coherent sheaf of dimension $d$ ,and let $c=n-d$ be its codimension.The dual sheafis defined as:
$$E^D=\mathcal{Ext}_{X}^{c}(E,\omega_{X})$$
It is then stated that this definition of the dual has the advantage of being independent of the ambient space
Namely if $X,Y$ are smooth, $i:X \rightarrow Y$ is a closed embedding, then: $$(i_{*}E)^D=i_{*}(E^D)$$ Can you give some advice to prove this equality? Thanks a lot.