Definition of a quotient group in Dummit-Foote

I'm reading the section on quotient groups in Dummit and Foote, and they give somewhat non-standard definition of a quotient group. I was wondering whether there is an easy way to see right away for someone who is familiar with the standard definition of a quotient group that DF's definition is equivalent to the standard one?

What I can see. The standard definition defines $$G/K$$ (as a set) as the set of left cosets of $$K$$ in $$G$$. Every such coset is an equivalence class under the equivalence relation on $$G$$ given by $$g_1\sim g_2\iff g_1=g_2k$$ for some $$k\in K$$. The fibers of $$\phi:G\to H$$ are the equivalence classes on $$G$$ given by $$g_1\sim g_2\iff \phi(g_1)=\phi(g_2)$$. So there are two partitions of $$G$$, and I think I need to see (1) why they are the same, (2) why multiplication is "the same". • I don't see why you think you have two partitions of $G$? – Bernard Sep 29 '18 at 20:55

Note that if $$g_1 = g_2k$$, then $$\varphi(g_1) = \varphi(g_2)$$ since $$\varphi(k) = e$$. Thus elements in the same coset are in the same fiber of $$\varphi$$.
If $$g_1$$ and $$g_2$$ are in the same fiber of $$\varphi$$, then $$\varphi(g_1) = \varphi(g_2)$$, so that $$g_1g_2^{-1}\in K$$, or $$g_1 = g_2k$$ for some $$k\in K$$.