The $C^\infty(M)$-linearity of the curvature implies that it is defined pointwise

Question

In the second paragraph of this passage from Loring Tu's book:

I don't understand why does the $C^\infty(M)$-linearity of the curvature implies that it is defined pointwise ? to be more precise I don't understand why it wouldn't be defined pointwise if it was not $C^\infty(M)$-linear ?