My lecture notes on differential geometry read the following (without proof):
For $M$ a manifold, let $TM = \bigcup_{p \in M} T_p M$ be the (disjoint) union of all its tangent spaces. Then, there exists a unique topology and smooth structure on $TM$ making it into a smooth manifold such that any section $X:M \rightarrow TM$ of the canonical projection map is smooth if and only if for all smooth functions $f$ on $M$, the function $Xf$ is smooth.
I am not quite sure as to how to approach this (I am talking about the uniqueness part, the usual topology and smooth structure evidently imply the desired equivalence). I found a related post without an answer here: Why is the manifold structure on the tangent bundle unique? It seems that in this post, OP also assumes the projection map to be continuous, whereas my lecture notes do not.
Any help would be appreciated.