**In short:** The question is how to go from the first equation on page 8, of this paper to the second equation.

**Some background**
I'm working in optimization and I am currently reading a paper
see page 8,Proof of Proposition 2.3. where the author uses a somewhat geometric notation which makes it a bit hard for me to grasp:

We are studying an equation $$f'(t) = Tf(t) + V(f(t))=H(f(t))$$ with initial value $f(0).$ Thus, the equation is linear in $T$ and in general nonlinear in $V.$

The idea is to split this up into $$f'_1(t) = Tf_1$$ and $$f'_2(t) = V(f_2(t))$$ because the variation of constant formula teaches us that (now I use the exponential map $e^{tH}(F)f(0) := F(f(t))$, and analogously for $e^{tT}(F)f_1(0) := F(f_1(t))$)

$$e^{tH}(id)f(0) = e^{tT}(id)f(0) + \int_0^t e^{(t-s)H}(D_V e^{sT}(id))(f(0)) ds$$ where $D_V$ is the Lie-derivative with respect to vector field $V$. You find this variation of constant formula in the above paper. The author then says, that since this equation determines $exp(tH)$ we can apply it again for the exponential $exp((t-s)H)$ appearing in the integrand. You see this step carried out by the author in the above paper.

However, I do not see why this is true, as this equation above only determines $e^{tH}(id)$ and not $e^{(t-s)H}(D_V e^{sT}(id))$ which appears in the integrand.

Can anybody shed some light on the arguments of the author of this paper?