Let $V$ be a vector space over a field $K$. Denote the dual of $V$ by $V^{*}$, that is $V^*=Hom_K(V,K)$. Suppose there is a morphism $\alpha: V \rightarrow V$. Then we know $\alpha$ induces a morphism $\alpha^*: V^* \rightarrow V^*$ defined as follows: $$\langle \alpha^*(f), x \rangle=\langle f, \alpha(x) \rangle$$ for $f \in V^*$ and $x \in V$.

I want to know if $V$ is a super vector space and $\alpha$ is a morphism of $V$, what is the usual form of $\alpha^*$? Do we still define $\langle \alpha^*(f),x \rangle= \langle f, \alpha(x) \rangle$? Or $\langle \alpha^*(f),x \rangle=(-1)^{|f||x|} \langle f, \alpha \rangle$? Or other forms?

Here is the definition of super vector space:https://en.wikipedia.org/wiki/Super_vector_space. Thank you for your help.


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