1. $Hom(K,Hom(M,E))\cong Hom(K\otimes M,E)$  
 
 2.  $F$ is flat iff $F\otimes -$ is exact.  
 3. Let $E$ be [injective  cogenerator](https://math.stackexchange.com/questions/229168/injective-cogenerators-in-the-category-of-modules-over-a-noetherian-ring). Then $0 \longrightarrow X \longrightarrow Y \longrightarrow Z \longrightarrow 0$ is exact iff $0 \longrightarrow Hom(Z, E) \longrightarrow Hom(Y, E) \longrightarrow Hom(X, E) \longrightarrow 0$ is exact.    
***
Using (1) and (2), you can see that If $M$ is flat then $Hom(m,E)$ is injective (as abx said). Using (3), you can see that if $E$ is injective  cogenerator, then you have also the necessary condition.