Proving that Tensor Product is Associative

Solution 1:

Here's a sketch of the argument based on the hint from Stefan Walter.

Start by fixing $x\in X$, and define the bilinear map $Y\times Z\to (X\otimes Y)\otimes Z$ by $$(y,z)\mapsto (x\otimes y)\otimes z$$

By the universal property stated in my question, this induces a linear map $$A_{x}:Y\otimes Z\to (X\otimes Y)\otimes Z\text{ such that }A_{x}(y\otimes z) = (x\otimes y)\otimes z\text{ for every }y\in Y,z\in Z$$

Next define the bilinear map $X\times (Y\otimes Z)\to (X\otimes Y)\otimes Z$ by $$(x,\sum_{i=1}^{n}y_{i}\otimes z_{i})\mapsto A_{x}(\sum_{i=1}^{n}y_{i}\otimes z_{i})$$

Passing this map through the universal property yields the isomorphism.