= Solution
The coordinate transformation law for a <affine connection> has a homogeneous term with one upper and two lower index factors, and an inhomogeneous term involving second coordinate derivatives. The latter is the same for both <Levi-Civita connections>, so it cancels in their difference. Consequently
$$
\boxed{T'{}^a{}_{bc}=\frac{\partial x'^a}{\partial x^d}\frac{\partial x^e}{\partial x'^b}\frac{\partial x^f}{\partial x'^c}T^d{}_{ef},}
$$
which is exactly the <tensor component transformation law>. Equivalently, $(\widetilde\nabla-\nabla)_XY$ is linear over functions in both $X$ and $Y$.
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