ScalingStacks

Proof. For any $K$-local object $X$ of $\pre(\mathcal{R})$, a morphism $Y\to X$ represents an object of $(K^{-1}\pre(\mathcal{R}))_{/X}$ if and only if, for any morphism $U\to V$ of $K_{0}$, the square

is homotopy cartesian, since the horizontal map at the bottom is an equivalence. For this, it suffices to show that the induced map on homotopy fibers over any vertex of $\map(V,X)$ is an equivalence. Unpacking this, we obtain the condition that for any morphism $V\to X$, the map

$$\map_{/X}(V,Y)\to\map_{/X}(U,Y)$$

is a weak equivalence. We therefore deduce that $(K^{-1}\pre(\mathcal{R}))_{/X}$ may be exhibited as a localization $K_{X}^{-1}(\pre(\mathcal{R})_{/X})$, where $K_{X}$ is the strongly saturated class generated by the set of diagrams of the form

in which $\phi\in K_{0}$.

Now suppose $\eta\colon Z\to C_{k}$ a morphism of $K^{-1}\pre(\mathcal{R})$. Since colimits are universal in $\pre(\mathcal{R})$ [28, § 6.1.1], the functor

$$\pre(\mathcal{R})_{/C_{k}}\to\pre(\mathcal{R})_{/Z}$$

given by pullback along $\eta$ preserves all colimits, and the universal property of localizations guarantees that the composite

$$\pre(\mathcal{R})_{/C_{k}}\to\pre(\mathcal{R})_{/Z}\to K_{Z}^{-1}(\pre(\mathcal{R})_{/Z})\simeq(K^{-1}\pre(\mathcal{R}))_{/Z}$$

descends to a colimit-preserving functor

$$(K^{-1}\pre(\mathcal{R}))_{/C_{k}}\simeq K_{C_{k}}^{-1}(\pre(\mathcal{R})_{/C_{k}})\to K_{Z}^{-1}(\pre(\mathcal{R})_{/Z})\simeq(K^{-1}\pre(\mathcal{R}))_{/Z}$$

(which then must also be given by the pullback along $\eta$) if and only if, for any diagram

in which $0\leq k\leq n$ and $\phi\in K_{0}$, the induced morphism $U\times_{C_{k}}Z\to V\times_{C_{k}}Z$ lies in $K$.

It is clear that it suffices to check this only for nondegenerate morphisms $V\to C_{k}$. It now remains only to show that it suffices to check this for objects $Z$ among the essential image of $\mathcal{R}$. This follows from the fact that the class $K$ is strongly saturated and the fact that $\mathcal{R}$ generates $\pre(\mathcal{R})$ under colimits. ∎