I'm trying to find the covariant derivative of a covariant derivative, i.e. $\nabla_a (\nabla_b V_c)$.
This is something I've taken for granted a lot in calculations, namely I though that by the Leibniz rule we just have:
$$\nabla_a (\nabla_b V_c) = \partial_a(\nabla_b V_c) - \Gamma_{ab}^{d}\nabla_c V_d - \Gamma_{ac}^{d} \nabla_d V_c$$
However when we prove that the covariant derivative of a $(0,2)$ tensor is the above, we use the fact that the covariant derivative satisfies a Leibniz rule on $(0,1)$ tensors: $\nabla_a(w_b v_c) = v_c\nabla_a(w_b) + w_b\nabla_a(v_c)$. However $\nabla_a$ on it's own is not a tensor so how do we have the above formula for it's covariant derivative?