Let $M$ and $N$ be smooth manifolds, let $f: M \to N$ be a smooth map, and let $\pi_E: E \to N$ and $\pi_F: F \to N$ be smooth vector bundles. Then there are canonical smooth vector bundle isomorphisms over $M$
The direct-sum isomorphism is the unique bundle map whose fibre map over $x \in M$ sends
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\begin{align*}
(x, e \oplus u) \mapsto (x,e) \oplus (x,u),
\end{align*}
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for $e \in E_{f(x)}$ and $u \in F_{f(x)}$. The tensor-product isomorphism is the unique bundle map whose fibre map over $x \in M$ sends pure tensors by
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\begin{align*}
(x, e \otimes u) \mapsto (x,e) \otimes (x,u),
\end{align*}
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for $e \in E_{f(x)}$ and $u \in F_{f(x)}$, and extends linearly to $(E \otimes F)_{f(x)}$.