(Third Isomorphism Theorem). If $H$ and $K$ are normal subgroups of a group $G$ such that $K\lt H$, then $H/K$ is a normal subgroup of $G/K$ and $(G/K)/(H/K)\cong G/H$.
Hungerford’s proof: The identity map $1_G:G\to G$ has $1_G(K)\lt H$ and therefore induces an epimorphism $I: G/K \to G/H$, with $I(aK)=aH$. Since $H = I(aK)$ if and only if $a\in H$, $\text{ker} I= \{aK \mid a \in H\}= H/K$. Hence $H/K \lhd G/K$ by Theorem 5.5 and $G/H = \text{Im} I\cong (G/K)/\text{ker}I = (G/K)/(H/K)$ by Corollary 5.7.
Question: I don’t understand following sentence in above proof: The identity map $1_G:G\to G$ has $1_G(K)\lt H$ and therefore induces an epimorphism $I: G/K \to G/H$, with $I(aK)=aH$. To be precise, use of identity map $1_G$ to define an epimorphism $I$. What is the purpose of identity map $1_G$?
I can show $I$ is well defined and epimorphism independent of identity map $1_G$.
Edit: As suggested by Professor Arturo Magidin I don’t need to rewrite proof of Theorem 6 Corollary 2 Section 5.