Topics in Algebra, a book written by Herstein, said the following thing on the first isomorphism theory:
[The book says] Theorem 2.7.1 is important, for it tells us precisely what groups can be expected to arise as homomorphic images of a given group. These must be expressible in the form $G/K$, where $K$ is normal in $G$. But, by Lemma 2.7.1, for any normal subgroup $N$ of $G$, $G/N$ is a homomorphic image of $G$. Thus there is a one-to-one correspondence between homomorphic images of $G$ and normal subgroups of $G$.
(In that book, Theorem 2.7.1 refers to the first isomorphism theorem, and Lemma 2.7.1 says for any normal subgroup N of G, G/N is a homomorphic image of G)
Would you check my following interpretation of Herstein's sentence?
[My interpretation] Let $\Omega=\{N|N\triangleleft G\}$ and $\Pi=\mathrm{Hom}(G)/\cong$, where $\mathrm{Hom}(G)$ denotes collection of homomorphism from $G$ to somewhere and $\cong$ denotes isomorphism relation of groups. Then, there exists one-to-one correspondence $\phi:\Omega\ni N\mapsto \{f\in\mathrm{Hom}(G)\,|\,f(G)\cong G/N\}\in\Pi$.
I found a similar question but I couldn't understand this. I am just a beginner of abstract math, so I'm sorry in advance if I use some symbols and notations strangely.