Theorem 3.3.14 [Bruns and Herzog, CMR]: Let $(R,m)$ be a CM local ring and $(R,m) \rightarrow (S,n)$ a flat local homomorphism. Then
(a) If $\omega_R$ exists (this is the canonical module of $R$) and $S/mS$ is Gorenstein, then $\omega_S = \omega_R \otimes_R S$. (b) If $C$ is a finite $R$-module and in addition $C$ is CM with $\omega_S = C \otimes_R S$, then $S/mS$ is Gorenstein and $\omega_R \cong C$.
Corollary 3.3.15 [Bruns and Herzog, CMR]: Let $(R,m) \rightarrow (S,n)$ be a homomorphism of local Noetherian rings. Then $S$ is Gorenstein if and only if $R, S/mS$ are Gorenstein.
Question: I can see the direction $R, S/mS$ are Gorenstein $\Rightarrow S$ is Gorenstein, but i don't see how the reverse direction follows from the Theorem.
In particular, if $S$ is Gorenstein then it is CM and so both $R,S/mS$ are CM. Also, by Theorem 3.3.7 $\omega_S$ exists. But then to apply the theorem we would have to show that $\omega_S = C \otimes S$ for some $R$-finite module $C$, which i don't see why is true.
I don't know if this follows from the theorem, but certainly follows from Proposition 1.2.16, Theorem 2.1.7, and Theorem 3.2.10 (from the same book).