Technical Example: Tensors This example is intended to be familiar to ML folks. Given any vector space V, we can define a category V–– with
Objects: all vector spaces that are isomorphic to V
Morphisms: all linear isomorphisms
Fix some other vector space W. Then taking tensor product with W defines a functorV––→V⊗W––––––––. The fact that this is a functor is equivalent to the oft-stated maxim that a tensor is something that transforms like a tensor. ‘Transforms like a tensor’, means that the tensor product transforms changes in a consistent way when you change your representation of the vector space V (that is, change basis).
I found this explanation a bit sparse.