I'm reading a lecture note about tensors, following is a proposition:
For $R$-module $M$ and $N$, there is a linear map $M^*\otimes_RN\rightarrow \text{Hom}_R(M,N)$ sending each elements tensor $\varphi\otimes n$ in $M^*\otimes_RN$ to the linear map $M\rightarrow N$ defined by $m\mapsto \varphi(m)n$.
I think it is obvious and argue like this:
We have a bilinear map from $M^*\times N$ to $\text{Hom}_R(M,N)$ defined by $(\varphi,n)\mapsto \varphi(\cdot)n$. So by universal property of tensors, we get a linear map from $M^*\otimes_RN$ to $\text{Hom}_R(M,N)$.
However, the author of the note does not prove like this and argues seems more complicated:
The function $M^*\times M\times N\rightarrow N$ given by $(\varphi,m,n)\mapsto \varphi(m)n$ is trilinear, so (by a mentioned theorem) this trilinear map induces a bilinear map $B:(M^*\otimes_RN)\times M\rightarrow N$ where $B(\varphi\otimes m,n)=\varphi(m)n$. For fixed $t\in M^*\otimes_RN$, $B(t,\cdot)$ is in $\text{Hom}_R(M,N)$, so we have a linear map $f:M^*\times N\rightarrow\text{Hom}_R(M,N)$ defined by $t\mapsto B(t,\cdot)$.
I wonder why the author argued like this? Is there any problem in my simple proof?
