I started self-learning category theory where I encountered the universal property of tensor products. Then there are a few problems that I have no ideas on how to get started on:
Question: Let $U,V,W$ be vector spaces, use universal property to show that $U\otimes(V\otimes W)\cong(U\otimes V)\otimes W$ and $U\otimes V\cong V\otimes U.$
I learnt a few ways of constructing tensor products, which can help proving these statements but I want to use the universal property instead.
Universal Property: Let $U$, $V$ be two vector spaces over the same field then there exists a unique vector space $U\otimes V$ and a bilinear map $b:U\times V\to U\otimes V$ such that for any bilinear map $f:U\times V\to X$, where $X$ is a vector space, there exists a linear map $\tilde f:U\otimes V\to X$ such that $f=\tilde f b.$
My some initial thoughts: Say we focus on $U\otimes(V\otimes W)\cong(U\otimes V)\otimes W$ first. To construct an isomorphism, we can find linear maps going in both directions. To get such linear map, we can use Universal Property, this means that we need to use $U\times (V\otimes W)$ and $(U\otimes V)\times W$ instead? But I am not really sure how to construct a bilinear map and indeed I am not sure if I am on the right path also.
Thank you so much in advance!!