Another version of correspondence theorem.

Question

Let $X,Y$ be a sets and $\sim$ is an equivalence relation on $X$. $X/\sim$ is the quotient set or partition by $\sim$. Let us consider the sets $\{\tilde f:X/\sim\to Y\}$ and $\{f:X\to Y:f$ has the property $*\}$, where $*$ is the property that $[x]=[y]\implies f(x)=f(y)$. Then we can see that there exists a bijection between these two sets. Using this bijection, for two groups $G$ and $G'$ and for a normal subgroup $H$ of $G$, we can define a bijection between $\{f:G\to G':f $is a homomorphism and $ H\subset \ker(f)\}$ and $\{\tilde f:G/H\to G': \tilde f$ is a homomorphism$\}$, I want to know is this another version of correspondence theorem?I also want to know what are the applications of this result. Can someone help me?