B is a flat A-module, and M is a finitely generated $A$-module then $σ:\operatorname{Hom}_A(M, N)⊗_AB→\operatorname{Hom}_B(M⊗_AB,N⊗_AB)$ is injective 
Let A and B be rings, and let f : A → B be a homomorphism
  of rings; we consider B as an A-module with the structure
  induced by f. Let M and N be two A-modules. There is a map of $B$-modules
  $σ : \operatorname{Hom}_A(M, N) ⊗_A B → \operatorname{Hom}_B(M ⊗_A B, N ⊗_A B)$. If B is a flat A-module, and $M$ is a finitely generated $A$-module, then σ is injective.

I have no idea on construct the map $\sigma$ and prove the claim. Can anyone just give me some hints? Thank you in advance !
 A: Notice that 
\begin{align*}
Hom_B\left(M \otimes_A B, N\otimes_A B\right)  \overset{f} \simeq 
Hom_A\left(M, Hom_B \left(B, N\otimes_A B \right)\right) 
\end{align*}
via Hom-Tensor Adjoint Property.
Now, \begin{align*}
Hom_A\left(M, Hom_B \left(B, N\otimes_A B \right)\right)  \overset{g}  \simeq Hom_A\left(M, N\otimes_A B\right)
\end{align*}
Observe that the diagram below is commutative -
$$\require{AMScd}
\begin{CD}
Hom_A\left(M,N\right)\otimes_A B @>{\sigma}>> Hom_B\left(M \otimes_A B, N\otimes_A B\right)\\
@V{\alpha}VV @V{g \circ f}VV \\
Hom_A\left(M, N\otimes_A B\right)@>{Id}>>Hom_A\left(M, N\otimes_A B\right)
\end{CD}$$
Since $Id$ and $g \circ f$ are isomorphisms, hence, it is enough to prove that the map $\alpha$ is injective.
As $M$ is a finitely generated $A$ module, let $F$ be a finite-rank free $A$-module that surjects onto $M$. We immediately obtain the exact sequence -
\begin{equation} 
0 \to Hom_A\left(M,N\right) \overset{i_0}\to Hom_A(F,N)
\end{equation} 
Tensoring by the flat $A$-module $B$, we get 
$$ 
0 \to Hom_A(M,N)\otimes_A B \overset{i}\to Hom_A(F,N)\otimes_A B
$$ 
Upon replacing $N$ with $N\otimes_A B$, we obtain the exact sequence -
$$ 
0 \to Hom_A\left(M,N\otimes_A B\right) \overset{j}\to Hom_A\left(F,N\otimes_A B\right)
$$ 
And yet again we have a commutative diagram -
$$\require{AMScd}
\begin{CD}
Hom_A\left(M,N\right)\otimes_A B @>{i}>> Hom_A(F,N)\otimes_A B\\
@V{\alpha}VV @V{h}VV \\
Hom_A\left(M,N\otimes_A B\right)@>{j}>>Hom_A\left(F,N\otimes_A B\right)
\end{CD}$$
The maps $i$ and $j$ are injective maps, and the map $h$ is an isomorphism. Therefore, the map $\alpha$ is injective as well.
