My textbook (Szekeres's A Course in Mathematical Physics) asks me the following set of questions (NB I am self-studying):
If $H$ is any subgroup of a group $G$, define the action of $G$ on $G/H$ by $\psi: G \to$ Sym($G/H$) with $\psi(g)(g'H) = gg'H$.
a) Show that this is always a transitive action of $G$ on $G/H$
b) Let $G$ have a transitive left action on a set $X$, and set $H = G_x$ to be the isotropy (stabilizer) group of any $x \in X$. Show that the map $i: G/H \to X$ defined by $i(gH) = gx$ is well-defined and bijective. [I note here that the transitive action alluded to is tacit in the notation $gx$. Call it $\varphi$ as required.]
c) Show that the left action of $G$ on $X$ can be identified with the action of $G$ on $G/H$ defined in (a).
Now I've done (a) and (b), but am a bit confused by (c). I assume it's alluding to there being some commutative diagram-type argument that I can make, saying something like "for every transitive action $\varphi$ of $G$ on $X$, we can think of it instead as $\varphi = i \circ \psi$, but I'm not sure if I'm missing something big. I really can't quite see how to start since the "identified with" is so vague here.