I'm reading the book Geometric Numerical Integration and there is a part I can't really grasp. Given an autonomous system $$ \dot y = f(y), \,y(0) = y_0 $$ with flow $\varphi_t(y_0) = y(t)$ we study numerical schemes $\Phi_h$ and define their time adjoint as $\Phi_h^* = \Phi_{-h}^{-1}$. If this scheme is such that $$ \Phi_h(y_0) = \varphi_h(y_0) + C(y_0) h^{p+1} + \mathcal O(h^{p+2}), $$ then we wish to prove that $$ \Phi_h^*(y_0) = \varphi_h(y_0) + (-1)^p C(y_0) h^{p+1} + \mathcal O(h^{p+2}). $$
Taking $-h$ and defining $y_1 = \Phi^*(y_0)$ we obtain that $$ \Phi_{-h}(y_1) = \varphi_{-h}(y_1) + C(y_1) (-h)^{p+1} + \mathcal O(h^{p+2}), $$ and also $$ \Phi_{-h}(\varphi_h(y_0)) = y_0 + C(\varphi_h(y_0)) (-h)^{p+1} + \mathcal O(h^{p+2}). $$ I can't see how to make that adjoint show up, so any help would be appreciated.
Pd: The authors write that the prove comes from noting that $\varphi_h(y_0) = y_0 + \mathcal O (h)$ and that the errors are such that $e = (I + \mathcal O(h))e^* $, but they are not defined explicitly so it's not so clear.
Best regards