I don't actually agree with the author that it's less confusing, but I'll abbreviate my other comment to answer your direct question. A tensor is a multilinear

I don't actually agree with the author that it's less confusing, but I'll abbreviate my other comment to answer your direct question. A tensor is a multilinear map from a product of n vector spaces to another vector space. Every tensor is an element of a tensor product, which is (loosely speaking) a vector space you obtain by multiplying the dimension of two other vector spaces. So for example, the real space R^nk is the tensor product obtained from combining R^n and R^k. To be more explicit, what you're doing is obtaining a new n by k matrix for every pair of vectors of R^n and R^k. In other words, take the Cartesian product of R^n and R^k. Then take every possible pair of vectors of the Cartesian product and combine them to form an n by k matrix. Now you have a tensor product, and each individual element of your tensor product is a tensor. Going back to the abstract: a monoidal category is a category C of objects equipped with a functor defined ⊗: * C x C -> C*. If you take the category of vector spaces or the category of modules (like vector spaces, but defined over rings instead of fields) and define your monoidal functor to be the tensor product, you turn them into tensor categories. Monoidal categories actually sit at a higher level of abstraction than tensor products, so I don't agree this makes monoidal categories less confusing. In fact it diminishes the abstraction of monoidal categories by forcing you to think of the monoidal operation as the tensor product instead of something which is more general. On the other hand if you're familiar with (and prefer) the computationally-oriented definition of tensors as multidimensional arrays, this framing of monoidal categories might ground them somewhat so they're more accessible.