Let $(E,M)$ be a vector bundle, $\phi$ an endomorphisms and $X$ a vector fields of $M$. I call $\nabla_{X\otimes \phi}$ a double connection if:

$$\nabla : E \rightarrow TM^*\otimes End(E)^* \otimes E$$

$$\nabla_{X \otimes \phi} (s+s')=\nabla_{X\otimes \phi} (s)+\nabla_{X\otimes \phi} (s')$$

$$\nabla_{X \otimes \phi}(f.s)= X (f) .\phi (s)+f.\nabla_{X \otimes \phi} (s)$$

With $s,s'$ two sections of $E$, and $f$ a smooth function of $M$.

What is the space of double connections?