The universal property of the Liouville $1$-form @MartinBrandenburg, given a smooth map $X\to Y$, I know how to construct its cotangent lift $T^\ast Y\to T^\ast X$, only when $X\to Y$ is a local diffeomorphism, but not in general. So we can work at least with the subcategory of local diffeomorphisms $X\to Y$.