Explain why the companion \$$\widehat{\mathrm{id}}\$$ of \$$\mathrm{id}:\mathcal{P}\to\mathcal{P}\$$ really has the formula given in Eq. \eqref{eq1}.

$$\label{eq1}\tag{4.25}U_{\mathcal{X}}(x,y):=\mathcal{X}(x,y).$$