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Suppose I have an endomorphism $J:TM \to TM$ and a connection on M. It is possible to define $\nabla_X J$ by transforming $J$ into a (1,1)-tensor and using the extension of $\nabla$ to tensors. Going back we get an endomorphism $\nabla_X J:TM \to TM$.

Is there a way to define $\nabla_X J:TM \to TM$ directly?

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    You're right. I was trying to say $\nabla_x J$. I'll correct it.2011-07-28

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As was mentioned in the comments, $\nabla L$ is not a (1,1) tensor. It is actually a (1,2) tensor. However contracting with a vector field $X$ gives us an endomorphism $\nabla_X L$ of $TM$ which is equal to $\nabla_X \circ L - L \circ \nabla_X$. You can check this is consistent with the definition you get when you extend $\nabla$ to $T^*M$ and then to $TM^* \otimes TM$.