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 $(E \otimes E',g \otimes g')$ .