Left exact functors preserve injective objects

Question

Let $G$ be a group and let $\operatorname{\textbf{Mod}}_G$ be the abelian category of $G$-modules, which has enough injectives.

Let $A\in \operatorname{Mod}_G$ and let $(-)^G$ the functor
\begin{equation}
\begin{aligned}
(-)^G: \operatorname{\textbf{Mod}}_G &\longrightarrow \operatorname{\textbf{Ab}}\\\\
A &\mapsto A^G:=\{a\in A: \forall g\in G,ga=a\}
\end{aligned}
\end{equation}

Let $A\longrightarrow I_{\bullet}$ be an injective resolution. Applying $(-)^G$, we get a (not right exact) resolution
\begin{equation}
0 \overset{}{\longrightarrow}A^G \overset{d_0}{\longrightarrow}I_0^G \overset{}{\longrightarrow} I_1^G\overset{d_1}{\longrightarrow}I_2^G\overset{d_2}{\longrightarrow}\dots
\end{equation}

Is it an injective resolution of $A^G$? I would say yes because we can apply Lemma 12.29.1. But the issue is that I don't know what does "injective maps" mean in that Lemma, since it doen't have a meaning unless we have a concrete category. Does "injective map" in that lemma mean "monomorphism"?

Thank you.

Answer 1

Yes, in the sense of Definition 12.5.3 and the subsequent Lemma 12.5.4.