I am reading "An introduction to manifolds" by Loring Tu. And there in the proof of the chain rule for differential of a map i couldn't understand the last step. The following is the step that i am unable to understand:
$$(G_*(F_*X_p))f=(F_*X_p)(f\circ G)$$
I am attaching the screen shot of proof where this is given and i have highlighted the step which i can't understand. Also note that they've defined the differential of a map in the previous section of which i am also attaching the screenshot.The only step that i can't understand is that is highlighted(in the screenshot) and written above.
Best Answer
By definition, $(G_*(Y_q))f=Y_q(f\circ G)$. Now take $Y_q=F_*(X_p)$