Question: Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a\_1,\dots,a\_n$. Then there is an isomorphism $\Phi: Hom\_R(A,B) \to Hom\_R(A,R)\otimes\_R B$ defined by $\Phi(\sigma)=\sum\_{i=1}^n \psi\_i\otimes \sigma\_i$, where $\sigma\_i=\sigma(a\_i)$ and $\psi\_i$ is the map such that $\psi\_i(a\_j)=\delta\_{ij}$. Is there another way to describe $\Phi$?

Answer: 
There was an École d'été sur la conjecture de Birch et Swinnerton-Dyer in Paris in the summer of 2002. The [videos](http://bsd-2002.institut.math.jussieu.fr/videos/) of the lectures (by P. Colmez, H. Darmon, B. Edixhoven, B. Mazur, J. Nekovár, J. Oesterlé, W. Stein, E. Urban and S. Zhang)
are available.
it seems that the lectures were not recorded (there are only photos on the conference website); 
so there is no point in searching for the videos.

However, there are a few video excerpts available from Professor Stein's channel..small consolation. 

(1) [bsd1](http://www.youtube.com/watch?v=Vo9yWAhBxAs) 

(2) [bsd2](http://www.youtube.com/watch?v=IIN2__t5Bsk) 

(3) [bsd3](http://www.youtube.com/watch?v=YcMgwfsP0u0)