Let $|X|$ denote the space $X$ with the discrete topology, and $i: |X| \to X$ denote the identity map. Then the usual $i^{-1} \dashv i_*$ adjunction gives us a monad $i_*i^{-1}$ on the category of sheaves on $X$, which takes any sheaf and produces a flabby (flasque) sheaf. The canonical unit map $1 \Rightarrow i_*i^{-1}$ inserts a sheaf into its "flabbification," and this is extended to produce the usual right resolution by flabby sheaves.
While doing some light introductory reading on bar constructions, I expected that the Godement resolution would be a prime example. Unfortunately (as far as I can tell) the bar construction constructs a simplicial object from an algebra over a monad (or dually a cosimplical object from a coalgebra over a comonad). But $A \to i_*i^{-1}A$ is obviously not an algebra for a monad, nor a coalgebra for a comonad.
Is the Godement right resolution by flabby sheaves indeed some sort of (co)bar construction? If so, how can we see that?