let $ X $ be a topological space and $ F $ a sheaf on $ X $. Let $ X_{\text{disc}} $ be the set $ X $ but now with the discrete topology.
One possible way to associate to $ F $ the Godement sheaf $ \operatorname{Gode} $(the "Godement construction") is given as follows:
Let $ p: X_{\text{disc}} \to X $ be the obviously continuous map induced by the identity. It induces adjoint direct and inverse image functors $ p_{*} $ and $ p^{-1} $. Then $ \operatorname{Gode}(F) := p_{*}\circ p^{-1}(F) $ . It's known that $ p_{*} $ and $ p^{-1} $ are adjoint functors and therefore the above functor $ p_{*}\circ p^{-1} $ regarded as endofunctor on the the category of sheaves on $ X $ induces an associated monad on the category of sheaves on $ X $ .
Next there is claimed that using this monad there is a way to turn a sheaf $ F $ into a coaugmented cosimplicial sheaf.
Question: how this construction turning $ F $ into a coaugmented cosimplicial sheaf works in detail?