Let $\mathscr{C}$ be an abelian category with enough injectives. Let $\mathscr{C}_A$ denote the abelian category of directed systems of objects of $\mathscr{C}$ indexed by $A$. Does $\mathscr{C}_A$ have enough injectives?
If $\{C_\alpha\}$ is an object of $\mathscr{C}_A$, then for each $\alpha \in A$, there is an injective object $I_\alpha$ and a monomorphism $C_\alpha \to I_\alpha$. One might hope that there is a monomorphism $\{C_\alpha\} \to \{I_\alpha\}$, but $\{I_\alpha\}$ does not appear to even be an object of $\mathscr{C}_A$: if $\alpha \leq \beta \leq \gamma$ in $A$, by using the definition of injective object and that $\{C_\alpha\}$ is a directed system, we do get morphisms $f_{\alpha\beta}:I_\alpha \to I_\beta$, $f_{\beta\gamma}:I_\beta \to I_\gamma$, and $f_{\alpha\gamma}:I_\alpha \to I_\gamma$, but these are not unique, so we cannot expect that $f_{\beta\gamma}f_{\alpha\beta}=f_{\alpha\gamma}$.
What might be another approach?