Let $Q$ and $Q'$ be smooth manifolds, let $f:Q\to Q'$ be a diffeomorphism, and let $\tau_Q:T^*Q\to Q$ and $\tau_{Q'}:T^*Q'\to Q'$ be the cotangent bundle projections. Let $T^*f:T^*Q'\to T^*Q$ be the contravariant cotangent lift, defined by sending each covector $\alpha\in T_{q'}^*Q'$ to the covector $T^*f(\alpha)\in T_{f^{-1}(q')}^*Q$ given by

\begin{align*} T^*f(\alpha)=\alpha\circ df_{f^{-1}(q')}. \end{align*}

Let $\lambda_Q\in\Omega^1(T^*Q)$ and $\lambda_{Q'}\in\Omega^1(T^*Q')$ be the tautological one-forms, meaning

\begin{align*} (\lambda_Q)_\beta(\xi)=\beta(d(\tau_Q)_\beta(\xi)) \end{align*}

for every $\beta\in T^*Q$ and $\xi\in T_\beta(T^*Q)$, and analogously for $Q'$. Then

\begin{align*} (T^*f)^*\lambda_Q=\lambda_{Q'}. \end{align*}

If the canonical symplectic forms are defined by $\omega_Q=-d\lambda_Q$ and $\omega_{Q'}=-d\lambda_{Q'}$, then

\begin{align*} (T^*f)^*\omega_Q=\omega_{Q'}. \end{align*}