My question concerns the proof of Proposition 4.2 in Bhatt-Mathew’s paper on the arc-topology, but my confusion is completely general and anyone familiar with limits in $\infty$-categories would know what to do. The situation is that they have a map of $R$-algebras $V\to \tilde V:=V_{\mathfrak p}\times V/\mathfrak p$ which is a covering map for their Grothendieck topology, and they consider an $\infty$-category $\mathcal C$ which is compactly generated by cotruncated objects and a functor $\mathcal F:R\mathrm{-alg}\to\mathcal C$ which preserves finite products ($V$ is a valuation ring with prime ideal $\mathfrak p$ and residue field $\kappa(\mathfrak p)$, but this is not important). They then claim that the totalisation $\lim\mathcal F(\tilde V^{\otimes\bullet+1})$ coincides with the pullback $\mathcal F(V_{\mathfrak p})\times_{\mathcal F(\kappa(\mathfrak p))}\mathcal F(V/\mathfrak p)$. Here, $\tilde V^{\otimes\bullet+1}$ means the cosimplicial $R$-algebra $\tilde V\rightrightarrows\tilde V\otimes_V\tilde V\substack{\rightarrow\\[-1em] \rightarrow \\[-1em] \rightarrow}\ldots$. It’s important that in this situation $V_{\mathfrak p}\otimes_V V_{\mathfrak p}=V_{\mathfrak p}$, $V/\mathfrak p\times_VV/\mathfrak p=V/\mathfrak p$, $V_{\mathfrak p}\otimes_VV/\mathfrak p=\kappa(\mathfrak p)$, and $\kappa(\mathfrak p)\otimes_VV_{\mathfrak p}=\kappa(\mathfrak p)=\kappa(\mathfrak p)\otimes_VV/\mathfrak p$. In fact, once you make these identifications, you can forget what $V$ and $\mathfrak p$ are and use the fact $\mathcal F$ preserves products to write down $\mathcal F(\tilde V^{\otimes\bullet+1})$ very concretely; the first few terms are $$\mathcal F(V_{\mathfrak p})\times \mathcal F(V/\mathfrak p)\rightrightarrows\mathcal F(V_{\mathfrak p})\times \mathcal F(V_{\mathfrak p}\otimes_V V/\mathfrak p)\times\mathcal F(V/\mathfrak p\otimes_VV_{\mathfrak p})\times\mathcal F(V/\mathfrak p)\substack{\rightarrow\\[-1em] \rightarrow \\[-1em] \rightarrow}\ldots$$ where both of the terms with a tensor product are canonically equivalent to $\mathcal F(\kappa(\mathfrak p))$ and all the maps are the canonical ones.

I’m having trouble seeing the claim that this limit coincides with the pullback. (If it were a diagram in a 1-category it would be completely obvious.) The claim that $\mathcal C$ is compactly generated by cotruncated objects means that it suffices to show the result after applying $\operatorname{Map}_{\mathcal C}(x,-)$ and in this way replace $\mathcal C$ by $\mathcal S_{\leq n}$, the full subcategory of $\mathcal S$ on $n$-truncated spaces. Then we can truncate the above diagram so it is indexed by $\Delta_{\leq n+2}$ and deal with a finite diagram. The two approaches I’ve tried are to compute the above (infinite) totalization as a homotopy limit in $\mathcal S$ and to break up the products and show some finite sub-diagram of the resulting diagram is coinitial (left cofinal) in this specific situation. More specifically, I’ve tried to show the diagram $D$ is coinitial, where $D$ has four objects $\mathcal F(V_{\mathfrak p})$, $\mathcal F(V/\mathfrak p)$, and two copies of $\mathcal F(\kappa(\mathfrak p))$ and four non-identity arrows: one arrow each from the first two objects to each copy of the second. However, this doesn’t appear to be true. On the other hand, I haven’t found a nice enough way to express the homotopy limit in $\mathcal S$ to allow me to make the desired identification. Any tips would be a big help, thank you.

**Edit** A simplified model: objects $X,Y,Z$ with maps $X\to Y$, $Z\to Y$ in an $\infty$-category with limits and a tensor product $\otimes$ which commutes with $\times$; we have $X\otimes Y=Y=Z\otimes Y$, $X\otimes Z=Y$, $X\otimes X=X$, $Y\otimes Y=Y$, $Z\otimes Z=Z$, and we consider the cosimplicial object $(X\times Z)^{\otimes\bullet+1}$ with first three terms
$$X\times Z\rightrightarrows X\times (X\otimes Z)\times (Z\otimes X)\times Z\substack{\rightarrow\\[-1em] \rightarrow \\[-1em] \rightarrow}X\times (X\otimes X\otimes Z)\times(X\otimes Z\otimes X)\times(Z\otimes X\otimes X)\times (Z\otimes Z\otimes X)\times(Z\otimes X\otimes Z)\times(X\otimes Z\otimes Z)\times Z,$$
i.e.
$$X\times Z\rightrightarrows X\times Y\times Y\times Z\substack{\rightarrow\\[-1em] \rightarrow \\[-1em] \rightarrow}X\times Y^{\times 6}\times Z.$$
Breaking apart the product, the three maps above all send $X\to X$, and then send $X=X\otimes X\to X\otimes X\otimes Z$, $X\otimes Z\otimes X$, or $Z\otimes X\otimes X$.

The claim is that the limit of the above diagram is $X\times_YZ$.

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