Let $\mathbf{C}$ be a cosimplicial space and let the realization tower be a collection of simplicial sets $Tot_s{C}$ fitting into a tower $ ... \to Tot_s{C} \to Tot_{s-1}{C}...$. I will now define the simplicial set $Tot_s{C}$:
Let $\Delta_{(s)}$ be the s-skeleton of the cosimplicial space $\Delta$. Thus $\Delta_{(s)}$ is the minimal cosimplicial space whose $i$'th simplicial sets $i \leq s$ are the simplicial sets $\Delta_i$.
Define $Tot_s{C}$ to be the function complex $Hom_{\text{Cosimplicial spaces}}(\Delta_{(s)}, \mathbf{C})$.
Why does an augmentation of $\mathbf{C}$,
i.e. a map of simplicial sets $X \xrightarrow{\eta} \mathbf{C}([0])$ such that the maps $d^1_0, d^0_0:C([0]) \to C([1])$ satisfy $d^1_0 \circ \eta=d^0_0 \circ \eta$,
even give a map from $X \to Tot_s{C}$, say for $s=0$?
My motivation for asking is that this is part of an an assertion on page 3 of a paper on the Eilenberg Moore spectral sequence. (The assertion that the paper gives is that there is an augmentation of the tower $Tot_s{C}$.)
For the record, $\Delta$ denotes the cosimplicial space that when applied to $[n]$ gives the simplicial set $\Delta_n=Hom( \cdot, [n])$.
I think it's something along the following lines.
First, $\operatorname{Tot}_0(\mathbf{C})$ is just $\mathbf{C}[0]$, and we can take the map $X \to \operatorname{Tot}_0(\mathbf{C})$ to be the given augmentation $X \to \mathbf{C}[0]$.
In general, an augmentation $\eta: X \to \mathbf{C}[0]$ extends uniquely by the cosimplicial identities to a map of cosimplicial spaces $\tilde{\eta}: cX \to \mathbf{C}$ where $cX$ is the constant (aka discrete) cosimplicial space. By postcomposition this gives a map $$X \cong \operatorname{Hom}_{cS}(\Delta_{(s)}, cX) \xrightarrow{\tilde{\eta}_*} \operatorname{Hom}_{cS}(\Delta_{(s)}, \mathbf{C}) = \operatorname{Tot}_s(\mathbf{C}).$$
By naturality, this is compatible with the cofibration $\Delta_{(s)} \to \Delta_{(s+1)}$, so this gives a compatible system of maps $\{X \to \operatorname{Tot}_s(\mathbf{C})\}$.