Let's take this group action definition, taken from a textbook I'm reading:
Given two groups $G$, $H$ and a homomorphism $\phi=G \mapsto \text{Aut}(H)$, we say G acts on H through $$h^\sigma=h^{\phi(\sigma)}, h \in H, \sigma \in G$$
Now I've read this definition is actually a right action, which is not highlighted in the original text. I was wondering why this is a right action and I'd like to turn the above definition into a left action as well, to see the difference, possibly.
Thank in advance.