I understand that the covariant derivative of a covariant vector is as follows $$\nabla_\mu V_\nu=\partial_\mu V_\nu-\Gamma^\lambda_{\mu\nu}V_\lambda$$ But for basis vectors (which are also covariant), why is it that $$\nabla_\mu \textbf{e}_\nu=\partial_\mu \textbf{e}_\nu+\Gamma^\lambda_{\mu\nu}\textbf{e}_\lambda$$ Is there a proof for this?
I also have another issue. Using the definition of Christoffel symbols, $$\partial_\mu \textbf{e}_\nu=\Gamma^\lambda_{\mu\nu}\textbf{e}_\lambda$$ substituting this to the covariant derivative, I get $$\nabla_\mu \textbf{e}_\nu=2\partial_\mu \textbf{e}_\nu=2\Gamma^\lambda_{\mu\nu}\textbf{e}_\lambda$$ What is going on here?