A cotangent vector is a linear functional of a tangent vector at point p, i.e. it’s a dual vector, by the way definition of function and functional is: A function takes numbers (or vectors) as inputs and produces numbers (or vectors) as outputs. A functional takes a function as input and produces a scalar (number) as output.

Momentum is cotangent vector to velocity.

how cotangent vector transform? they pull back. it’s called expansion of differentials;

To summarize: