First, I will briefly discuss the history of the conceptual infrastructure of mathematics. The last important contribution to this infrastructure is the introduction ,in the nineteen forties, of the concepts of category, functor, and natural transformation . Although Category Theory as a specialty has seen extensive development since then, its basic concepts are still used very little as part of the conceptual infrastructure of other branches of mathematics. I believe that this is a pity . I will introduce and explain the concept of a ~tensor functor~, and I will show how it can be used to gain better insight into the basic concepts of linear algebra and differential geometry.

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