I am trying to prove that the category of projective systems of $R$-modules indexed over a directed poset $I$ has enough injective objects.
As suggested in Jensen's book ("Les foncteurs dérivés de lim..."), given a projective system $\{M_i\}$ one should choose an injection $M_i \longrightarrow E_i$ with $E_i$ injective $R$-module, for any $i \in I$ and then set $F_i := \prod_{k \leq i} E_k$. The family $\{F_i:i \in I\}$ is a projective system of $R$-modules (with the natural projections) and has a monomorphism from the object $\{M_i\}$.
The problem is I am stuck in proving that the latter is indeed an injective object in the category of projective systems of $R$-modules. My attempt is the following. Let $\{N_i\} \longrightarrow \{L_i\}$ be a monomorphism of projective systems and $\{N_i\} \longrightarrow \{F_i\}$ be any morphism of projective systems. I need an extension $\{L_i\} \longrightarrow \{F_i\}$ of the latter. Fix $i \in I$; for any $k \leq i$ we have maps $g_{ki}:N_i \longrightarrow F_i \longrightarrow E_k$ (the outer map is the natural projection onto the factor $E_k$), and $E_k$ is injective, whence a map $h_{ki}:L_i \longrightarrow E_k$ extending $g_{ki}$. By universal property of the product we obtain a map $h_i:N_i \longrightarrow F_i$ which extends $g_i:N_i \longrightarrow F_i$ (looking at the pair of commutative diagrams defining $h_{ki}$ and $h_i$). This is OK, but now one should prove that the collection of maps $\{h_i\}$ is a map of projective systems, which I tried to and doesn't seem to be true. In my attempt I get to the point where, for two indices $i \leq j$, I should prove that $L_j \longrightarrow L_i \longrightarrow E_k$ is a commutative triangle for all $k \leq i$ (in other words, $h_{kj}=h_{ki} \circ (L_j \longrightarrow L_i)$), which doesn't seem to be the case in general. Also I noticed that I never used the fact that $\{N_i\} \longrightarrow \{L_i\}$ and $\{N_i\} \longrightarrow \{F_i\}$ are morphisms of projective systems.
Can anyone help me figure out the point?