If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie algébrique et théorie des faisceaux". For example, see here. I am interested in comonads in the category $\mathsf{Sh}(X)$ of sheaves of abelian groups on a topological space $X$, so I looked out this paper and attempted some translation. I think that the relevant part is, given a sheaf $\mathscr{F}$, we define a new sheaf $C \mathscr{F}$ where $$ C\mathscr{F}(U) = \{ \text{functions } s \colon U \longrightarrow \mathscr{E}_\mathscr{F} \mid s(x) \in \mathscr{F}_x \forall x \in U \} $$ where $\mathscr{E}_\mathscr{F}$ is the étalé space of $\mathscr{F}$, and $\mathscr{F}_x$ is the stalk of $\mathscr{F} $ at $x \in X$.
My question is, is this really a comonad? I have found it hard to define a natural transformation $\varepsilon \colon C \longrightarrow 1$ that would turn $C$ into a comonad. Is there another functor $\mathsf{Sh}(X) \longrightarrow \mathsf{Sh}(X)$ which is the comonad which is being referred to? Is the original statement incorrect?