First off, if is the identity map on a smooth manifold , then should be the identity map between the tangent bundles. That is, at each point we should have the identity map on this vector space. And indeed, if we let be any coordinate patch around we know that the matrix of with respect to these local coordinates is the Jacobian of the coordinate function . But this Jacobian is clearly the identity matrix, proving our claim.
More importantly, if and are two smooth maps, then their composition is also smooth. Given a point we can define the derivatives , , and . I say that . And since this holds at every point we can write , proving functoriality.
So, let’s take a vector and see what happens. Taking a test function we calculate
And so , just as we claimed.
We should note, here, how this recalls the Newtonian notation for the chain rule, where we wrote . Of course, multiplication is changed into composition of linear maps, but that little detail will be cleared up soon (if you don’t already see it).