I am currently trying to understand the proof of Prop 1.10 in the book presented in the title. I will leave a screen capture below of the part in question for reference. The only thing that I can't really why it is true and need help on, is the isomorphism of sheaves:
$$ \mathcal{Diff}(\Omega_{X}, \Omega_{X}) \cong \Omega_{X} \otimes D_{X} \otimes \Omega_{X}^{\otimes -1}$$
Here, tensor products are taken over $\mathcal{O}_{X}$, and $\Omega_{X}^{\otimes -1} = \mathcal{Hom}_{\mathcal{O}_{X}}(\Omega_{X} , \mathcal{O}_{X})$, which is an inverse of $\Omega_{X}$ with respect to $\otimes_{\mathcal{O}_{X}}$. $\mathcal{Diff}(\Omega_{X}, \Omega_{X})$ is defined in page 2 of the book, and is a subsheaf of $\mathcal{Hom}_{\mathbb{C}}(\Omega_{X}, \Omega_{X})$ where any section $f$ satisfies the following:
For any section $s \in \Omega_{X}$ and any $a \in \mathcal{O}_{X}$, there finite sections $s_{i} \in \Omega_{X}, P_{i} \in D_{X}$ with $f_{x}(a_{x} s_{x}) = \sum \limits_{i} P_{i}(a)_{x} (s_{i})_{x}$.
So, can anyone help me to figure out what the isomorphism written above is explicitely?
Thank you for all the help in advance :)
