If $G$ is a group, $H$ a subgroup, and $N$ a left $\mathbb{Z}[H]-$module, I've learned the following construction: $$\mathrm{coInd}_H^G(N) = \mathrm{Hom}_{\mathbb{Z}[H]}(\mathbb{Z}[G], N)$$ where $\mathbb{Z}[G]$ is given the structure of a left $\mathbb{Z}[H]$-module setting $$h\cdot x = xh^{-1}$$ for every $x\in G$ and $h\in H$. Then one makes $\mathrm{coInd}_H^G(N)$ a left $\mathbb{Z}[G]$-module declaring that for every $g\in G$ and $\varphi\in\mathrm{coInd}_H^G(N)$ $$(g\cdot\varphi)(x) = \varphi(g^{-1}x).$$
Is there some reason why we are taking all those inverses instead of just defining, in a way that looks more natural to me, the $H$-structure on $\mathbb{Z}[G]$ setting $h\cdot x = hx$ and then the $G-$structure on $\mathrm{coInd}_H^G(N)$ setting $(g\cdot\varphi)(x) = \varphi(xg)$?
I think the two definitions are equivalent: if $\varphi\in\mathrm{coInd}_H^G(N)$ one can define $\psi:\mathbb{Z}[G]\to N$ such that $\psi(x) = \varphi(x^{-1})$ for every $x\in G$. Now $\psi$ is an element of the coinduced module constructed following the second definition, and the map sending every $\varphi$ to the corresponding $\psi$ gives an isomorphism. Am I wrong? If not, why is the first definition preferred over the second?
I like your definition better, because it fits the general pattern "for an $S\mathrm{-}R\mathrm{-}$bimodule $_SM_R$ and a left $S$-module $_SN$, $\mathrm{Hom}_S(M, N)$ inherits left $R$-module structure from the right $R$-module structure of $M$".
I agree that the two definitions are equivalent. Perhaps the upshot of the first definition is that everything is defined in terms of left $H-$ and $G-$modules, without invoking right $G$-modules ($H$-modules, resp.).
But the fact that you can make this translation is not surprising, since the categories of left and right $G$-modules are equivalent for any group $G$: the equivalence is given precisely by this inverse of action operation, that is, a left $G$-module $_{\mathbb{Z}[G]}M$ corresponds to the right $G$-module $M_{\mathbb{Z}[G]},$ where the underlying Abelian group is the same, and the action is defined by $m\cdot g:=g^{-1}\cdot m$.