Suppose $TM\to M$ is the tengent bundle over the close Riemannian manifold $M$. Let $\nabla$ be the Levi-Civita connection, $S$ and $T$ are two $(1,1)$-tensor, i.e. at each point $x\in M$, we can view $S_x$ and $T_x$ as linear homomorphisms between $T_xM$ to itself.
For any vector $v\in T_xM$ one can compose this two tensors, i.e. $(S\circ T)(v)= S(T(v))$. Now my question is the composition $S\circ T$ should be $(1,1)$ tensor, and what is the covariant derivative of it?
My understand is $S \circ T$ is different with $S\otimes T$, right? (for the second one, I know the answer)