I am confused as to what "universal" means as used in my lecture course on Algebra in the context of $R$-bilinear maps where $R$ is a ring. The relevant section of my notes is:
Let $L, M, N, T$ be $R$-modules.
Suppose $\phi : M \times N \rightarrow T$ is $R$-bilinear, and $\theta : T \rightarrow L$ is linear.
Then $\theta \circ\phi: M \times N \rightarrow L$ is $R$-bilinear, so we get:
$\phi^* : \{\text{$R$-module maps } T \rightarrow L\} \rightarrow \{R\text{-bilinear maps } M \times N \rightarrow L\}$
Definition: $\phi$ is universal if $\phi^*$ is a one-to-one correspondence.
I am really confused as to what this means. What is $\phi^*$? Or rather, what is it doing? How does it depend on $\phi$? What does it mean for $\phi$ to be universal?
I would really appreciate some help understanding this, thank you.
The definition is a little bit sloppy. The role of $M, N, T$ is different from the role of $L$. You should see $M, N, T$ as "fixed": your definition will tell you something about a specific bilinear mapping $\phi: M \times N \to T$.
For each $L$, you have a mapping $\phi^*_L$ defined by $$ \phi^*_L(\theta) = \theta \circ \phi. $$ Check for yourself that if $\theta : T \to L$ is an $R$-module map, then $\theta \circ \phi: M \times N \to L$ is an $R$-bilinear mapping. Thus the definition of $\phi^*$ makes sense with the domain and codomain given in the question.
Now the definition should be: $\phi$ is universal if $\phi^*_L$ is a one-to-one correspondence for each $L$.
By definition it means what I wrote above.
How you work with it is roughly as follows. Suppose you have any $R$-bilinear mapping $\alpha: M \times N \to K$. You might want to say something about this mapping, but bilinear mappings are complicated and you don't understand them very well yet. The definition above says that $\alpha$ lies in the image of $\phi^*_K$, and in particular there is a unique $\theta: T \to K$ such that $\alpha = \phi^*_K(\theta) = \theta \circ \phi$. This means that to understand $\alpha$, you only have to understand $\phi$ -- which you typically have some kind of explicit construction for -- and $\theta$, which is an $R$-linear mapping and therefore quite possibly easier to understand.
In a sense, it tells you that a bilinear mapping $\alpha$ falls into two parts: a part $\phi$, which "takes care of the bilinearity", and a $\theta$, which "takes care" of the specific properties of the mapping $\alpha$. Note that $\theta$ depends on $\alpha$, but $\phi$ does not: in a sense $\phi$ is a blank template for all possible "bilinearity" of mappings out of $M \times N$. This might be why it is called "universal".