I know the definition of the tensor product, and I can somehow understand its importance, but among several constructions in mathematics, somehow I just never grasped the meaning of the tensor product. I don't understand them in the same sense as I understand the concepts of say products, coproducts, semidirect product, fibre product, pushout, kernel, cokernel and their universal properties.

When proving things I have a feeling for when I should take for instance the fibre product, and I have a feeling for when it is appropriate to consider certain structures, but the tensor product just never came naturally to me.

Just to make it clear. I understand the algebra behind the construction of the tensor product. I can operate with it. I can also verify and prove the Hom-Tensor adjunction, although I do not fully grasp what it really is telling me. I also have seen that the right-exactness of tensoring provides useful, but I never really understood what is going on.

Someone told me that I should look at tensor products as *linearizing*, but I am not sure in what context this was intended as a visualization.

I am wondering if anyone knows a reference or could explain to me what the tensor product really means. I am seeking examples where it is "obvious" that tensoring will help solve a problem, and how it solves the problem.

This is probably a very wide question, and I have not specified the context in which the tensor product is considered, but I feel like there must be some general explanation for what the tensor product really does.