Tensor product of metric connections is metric

Statement

A differential manifold $M$ . Two metric bundles $(E, g)$ and $(E^{'}, g^{'})$ (i.e., $E, E^{'}$ are vector bundles and $g, g^{'}$ are Riemannian metrics or pseudo-Riemannian metrics on these). $\nabla, \nabla^{'}$ are metric connections on $(E, g)$ and $(E^{'}, g^{'})$ respectively. Then, the tensor product of connections $\nabla \otimes \nabla^{'}$ is a metric connection on $(E \otimes E^{'}, g \otimes g^{'})$ .