$\newcommand\mb{\mathbb}\DeclareMathOperator{\Mor}{Mor}\DeclareMathOperator{\coInd}{coInd}$ Let $G$ be group, then we can define for an abelian group $N$ the co-induced module $\coInd^G(N) := \Mor_{\mb Z}(\mb Z[G], N)$ which is a $\mb Z[G]$-module. How is the $G$-module structure on $\coInd^G(N)$ defined? I can think of two ways, each one having its advantages.
- for $h \in G$, $f \in \Mor_{\mb Z}(\mb Z[G],N)$ we define $(hf)(g) := f(gh)$.
- for $h \in G, f \in \Mor_{\mb Z}(\mb Z[G],N)$ we define $(hf)(g) := f(h^{-1}g)$.
The first one has the advantage that if $M$ is a $G$-module, the unit map $\eta : M \to \DeclareMathOperator{\inclu}{Incl}\coInd^G(\inclu(M))$ which arises from the adjunction (here $\inclu : \mb Z[G]\text{-Mod} \to \operatorname{Ab}$ is the forgetful map) $$ \Mor_{\mb Z}(\inclu(M),N) = \Mor_{\mb Z[G]}(M,\coInd^G(N)) $$ which sends $\eta(m) := (g \mapsto gm)$ is a $\mb Z[G]$-module morphism: $\eta(hm) : (g \mapsto ghm) = h (\eta(m))$.
On the other hand, the second definition has the advantage that if $G$ is a finite group, the isomorphism between co-induction and induction (which is defined as $\DeclareMathOperator{\Ind}{Ind} \Ind^G(N) := N \otimes_{\mb Z} \mb Z[G]$) $$ \begin{aligned} \Mor_{\mb Z}(\mb Z[G],N) & \leftrightarrows N \otimes_{\mb Z}\mb Z[G] \\ \alpha : f & \mapsto \sum_{g \in G} f(g) \otimes g\\ 1_g \cdot n & \leftarrow\!\shortmid n \otimes g\\ \text{with } 1_g(h) & := \begin{cases} 1 & \text{if } h = g,\\ 0 & \text{if } h\neq g \end{cases} \end{aligned}$$ is an isomorphism of $\mb Z[G]$-modules: $$\alpha(hf) = \sum_{g \in G} (hf)(g) \otimes g = \sum_{g \in G}f(h^{-1}g) \otimes g = \sum_{g \in G} f(g) \otimes hg = h \alpha(f).$$
The first one $(hf)(g) := f(gh)$ is the correct one, as we can write the isomorphism between co-induction and induction instead as
$$ \begin{aligned} \DeclareMathOperator{\Mor}{Mor}\newcommand\mb{\mathbb} \Mor_{\mb Z}(\mb Z[G],N) & \leftrightarrows N \otimes_{\mb Z}\mb Z[G] \\ \alpha : f & \mapsto \sum_{g \in G} f(g^{-1}) \otimes g\\ 1_{g^{-1}} \cdot n & \leftarrow\!\shortmid n \otimes g\\ \text{with } 1_g(h) & := \begin{cases} 1 & \text{if } h = g,\\ 0 & \text{if } h\neq g \end{cases} \end{aligned} $$ with the property that $$ \alpha(hf) = \sum_{g \in G} (hf)(g) \otimes g = \sum_{g \in G}f(g^{-1}h) \otimes g = \sum_{g \in G} f(g^{-1}) \otimes hg = h \alpha(f). $$