Given a path $c: (-\epsilon,\epsilon)=I \to M$ in a manifold.
Define $\widetilde c:I \to TM$ (a kind of "lift") as the curve $t_0 \mapsto [c(t+t_0)] \in T_{c(t_0)}M$.
Is there a nice categorical definition of this "lift" using commutative diagrams in the smooth category?
My aim is to be able to say: $c: I \to M$ is an integral curve of $X$ iff $\widetilde c = Xoc$
There is no such "categorically defined lift," at least in the sense that it natural. This is an essential point in the definition of connection, the fact that there is no canonically defined lift for vetors to their tangent bundle. http://en.wikipedia.org/wiki/Ehresmann_connection
If you are in a Riemannian manifold, the situation is slightly less hopeless. There is a unique torsion free metric connection (the Levi-Civita connection.) You can then consider the lits defined by such connection.