I know that if $E$ and $F$ are two vector bundle with connection $\nabla^E$ and $\nabla^F$, then it is natural to define tensor connection $\nabla = \nabla^E\otimes 1 + 1 \otimes \nabla^F$ on $E \otimes F$.
By I have a stupid question: is is $\nabla^E \otimes 1$ also a connection?
It seems to satisfy the conditions: $\nabla(f\sigma_E\otimes \sigma_F) = \nabla^E(f\sigma_E)\otimes\sigma_F = (\nabla f) \sigma_E \otimes \sigma_F +f \nabla^E\sigma_E\otimes \sigma_F = \nabla f \sigma_E \otimes \sigma_F + f\nabla(\sigma_E \otimes \sigma_F)$.
I guess the reason is the above condition is true only if I put $f$ with $\sigma_E$; and this ambiguity is not desired. But I'm not sure.