Unity of Opposites

In https://unity-of-opposites.blogspot.com/2020/06/godement-part-iv-definition-of-monads.html we introduced the Godement monad. To recap, if $X$ is any topological space, and we form the disjoint union of points $X^\delta = \coprod_{x\in X} x$ equipped with the discrete topology, there is a canonical map $i :X^\delta\to X$ which is the identity on points. This gives rise to two adjoint functors between the corresponding sheaf categories, the direct image functor and the inverse image functor. Their composition gives rise to an endofunctor on $Sh(X)$; this endofunctor is a monad, $T$. Its unit is the unit of the adjunction. Given any sheaf $\mathcal{F}$, the sequence of sheaves $T^{n+1}(\mathcal{F})$ is naturally endowed with the structure of a cosimplicial object, with augmentation map $\mathcal{F}\to T(\mathcal{F})$. By purely formal abstract nonsense, (and all the more beautiful for how trivial it is) the resulting cochain complex given by taking alternating sums of the coface maps ...