# A question about the dual of super vector space

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.