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?