Question about limit of cosimplicial diagram associated with a sheaf Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the equalizer of $$\prod_iFU_i\rightrightarrows\prod_{i,j}FU_{ij}$$
is in fact the limit of the entire cosimplicial diagram (degeneracy maps omitted)
$$\prod_iFU_i\rightrightarrows\prod_{i,j}FU_{ij}\substack{\longrightarrow \\ \longrightarrow \\ \longrightarrow}\prod_{ijk}F(U_{ijk})\substack{\longrightarrow \\ \longrightarrow \\ \longrightarrow \\ \longrightarrow}\cdots$$
I'm trying to work this out using the concrete description of limits in the category of set, but I'm confused and stuck at the very beginning. I would like some guidance in this verification as well as geometric explanations of why we can truncate above level 1 and ignore degeneracy maps.
 A: Here's a very concrete argument. A limit of the cosimplicial object is an $X$ which comes a priori with maps to each of the cosimplicial levels, but since $0$ is weakly initial in $\Delta$, only needs to be given a map $f:X\to \prod FU_i$, from which the other maps are determined. Now $f$ is equalized by the two face maps, so $f$ factors through the equalizer $Y$ of the truncated diagram above. On the other hand, try to define a cone with tip $Y$ over the whole cosimplicial diagram starting with $g_0=g:Y\to \prod FU_i$ using, say, the zeroth face maps $d_0^i$. So we get maps $g_i$ to each level of the cosimplicial diagram. We need to show each face map composes with $g_i$ to get $g_{i+1}$. This happens by a double induction on $k$ and $i$ because of the cosimplicial identity 
$d_k^i \circ d_{k-1}^{i-1}=d_{k-1}^i\circ d_{k-1}^{i-1}$. Now the degeneracy maps are handled because $s^i_kg^i=s^i_kd^{i-1}_kg^{i-1}=g^{i-1}$, using the cosimplicial identities again and the cone over the face maps. Thus $g$ induces a cone of $Y$ over the cosimplicial diagram, yielding a factorization of $g$ through $f$. These two factorization so combine in the usual way to show $g$ displays $Y$ as a limit of the whole cosimplicial diagram.
All this was pure algebra of simplicial sets. Geometrically, the point is simply that the restriction of a section over $A$ to $A\cap B\cap C$ is the restriction of its restriction to $A\cap B$ or $A\cap C$, so if you have some sections which are consistent on twofold intersections, these last restrictions agree with the restriction of that over $B$ to $A\cap B, C$ to $A\cap C$ etc, so they're guaranteed to agree on threefold and higher intersections. In (higher) stack theory, the story no longer goes through: you choose isomorphisms between sections over twofold intersections, which must commute when restricted to threefold intersections, so you need a diagram truncated one level higher (or for higher stacks, you might need the entire diagram.)
