This is an exercise in Ana Canas Da silva's notes, section 2.3.
Let $X$ be a smooth manifold and $T^*X$ its cotangent bundle with projection $\pi:T^*M\to M$. For a coordinate chart $(U,x_1,\dots,x_n)$ for $X$, we can write any covector $v\in \pi^{-1}(U)$ as $v=\sum_i a^i dx_i$, and $(x_1,\dots,x_n,a^1,\dots,a^n)$ gives a local coordinate system on $\pi^{-1}(U)$. It can be seen that the 1-form $\alpha:=\sum_i a^i dx_i$ on $\pi^{-1}(U)$ does not depend on the chart $(U,x_1,\dots,x_n)$, so we get a global 1-form $\alpha$ on $T^*M$. This is called the tautological 1-form (or the Liouville 1-form) on $T^*M$.
I have to show that $\alpha$ is uniquely characterized by the property that, for every 1-form $\mu:X\to T^*X$, $\mu^*\alpha=\mu$. I have shown that $\alpha$ has this property. But how can we show that $\alpha$ is uniquely characterized by this property?