For an additive category $\mathcal C$ there is the notion of a Serre functor on $\mathcal C$, i.e. a an autoequivalence $S$ of $C$ such that there exist isomorphisms $$Hom(A, S(B)) \cong Hom(B, A)^*$$ natural in $A,B \in \mathcal C$.

If $\mathcal D$ is a full subcategory of $\mathcal C$ that is preserved by $S$, when is the restriction of $S$ a Serre functor of $\mathcal D$?

The particular case I am wondering about is the following: In the paper [1], the authors compute the Serre functor S of $D^b(\mathcal O_\lambda)$ of the derived category of an integral block of the BGG category $\mathcal O$. Let $\mathcal O^{\mathfrak p}_\lambda$ be the corresponding block of the parabolic category $\mathcal O$. They authors state that $D^b(\mathcal O^{\mathfrak p}_\lambda)$ is a full triangulated subcategory of $D^b(\mathcal O_\lambda)$ that is preserved by $S$. They then prove that the restriction of $S$ shifted by a non-zero integer $n$ is a Serre functor of $D^b(\mathcal O^{\mathfrak p}_\lambda)$.

Doesn't this imply that for complexes $A, B \in D^b(\mathcal O^{\mathfrak p}_\lambda)$ the above equality of Hom-spaces holds both for $S$ and $S[n]$?

[1] Volodymyr Mazorchuk and Catharina Stroppel. “Projective-injective modules,
Serre functors and symmetric algebras”. *Journal für die reine und angewandte Mathematik* 616 (2008), pp. 131–165.