Vector bundle morphism

Let $𝜋 : 𝐸 ↠ 𝑋$ and $𝜋' : 𝐸' ↠ 𝑋$ be vector bundles over a common base space $𝑋$ in a category $𝖢$ . A vector bundle morphism $𝑓 : 𝐸 \to 𝐸'$ is a map $𝑓 \in 𝖢 (𝐸, 𝐸')$ such that #m/def/geo/diff

commutes in $𝖢$ and the restriction $𝑓 ↾ 𝐸 𝑝 : 𝐸 𝑝 \to 𝐸' 𝑝$ to each fibre is an $ℝ$ -linear map. Thus being a vector bundle morphism is a stronger property than being a bundle map. Vector bundle morphisms are the morphisms of $𝖵 𝖾 𝖼 𝗍 𝖢 𝑋$ .

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