In proving the periodicity of Fredholm operator Fred$_0(H)$ $\cong$ Fred$_{-2}(H)$ for proving Bott periodicity Dan Freed in his notes commented that (proposition 12.46)
\begin{equation*} \begin{aligned} \text{Fred}_0(V^* \otimes H) &\longrightarrow \text{Fred}_{-2}(H) \subset \text{Fred}_0(V \otimes V^* \otimes H) \\ A &\longmapsto \text{id}_V \otimes A \end{aligned} \end{equation*} by the fact that the only central endomorphisms of $V$ are multiples of id$_V$, where $V$ is a $\mathbb{Z}_2$-graded complex vector space with dimension 1 for both the even and odd components.
I really don't understand how this fact can lead to the claimed homeomorphism. Could somebody clarify for me?