I was studying this proposition:
Let $A$ be a finite dimensional unital $F$-algebra, let $K/F$ be a field extension and $n\geq 1$. Then:
- $M_n(F)\otimes_F A \simeq M_n(A)$ is a $F$-algebra isomorphism.
- $M_n(F)\otimes_F K \simeq M_n(K)$ is a $K$-algebra isomorphism.
and in the proof I found the first sentence not so clear: The F-algebra homomorphisms $M_n(F)\rightarrow M_n(A),M \mapsto M$ and $A \rightarrow M_n (A), a \mapsto aI_n$ have commuting images, and therefore there is a unique F-algebra homomorphism $\phi : M_n(F) \otimes_F A \rightarrow M_n(A)$ satisfying $\phi(M\otimes a)=aM, \forall M\in M_n(F),a\in A$.
Then the proof continues, but I don't get why the two images commute and why this leads us to that conclusion. Can you give me any hint?