The Parallel transport direction in computing Jacobian #1126

In semi-explicit Euler integration, I am confused about the Jtransport part of calcDiff:

Let us take $F_u$ as an example. The code is computing $\frac{dx_{n+1}}{du_n} =\frac{\partial x_{n+1}}{\partial \delta x} \frac{\partial \delta x}{\partial u_n} $.
I understand the first part of the code is computing $\frac{\partial \delta x}{\partial u_n} $, and then call JintegrateTransport to do the transportation, i.e. left multiply $\frac{\partial x_{n+1}}{\partial \delta x} $:

state_->JintegrateTransport(x, d->dx, d->Fu, second); 

If I understand correctly, $\frac{\partial \delta x}{\partial u_n}$ has column vectors in tangent space at $x_n$. The desired quantity $\frac{dx_{n+1}}{du_n}$ has column vectors in tangent space at $x_{n+1}$.
Therefore, we are transporting vectors from tangent space at $x_n$ to that at $x_{n+1}=x_{n}\oplus\delta x$.

But in the documentation of JintegrateTransport, it says:

This function performs the parallel transportation of an input matrix whose columns are expressed in the tangent space at $x\oplus\delta x$ to the tangent space at $x$ point.

It is the reverse of my argument above.
What is wrong with my arguments?

Thanks!