ScalingStacks

Proposition 8.5. Let $\mathcal{R}$ be a small category, and let $i\colon\mathcal{R}\to\gaunt_{n}^{\omega}$ be a functor. Let $K$ be a strongly saturated class of morphisms in $\pre(\mathcal{R})$ of small generation. Denote by $i^{*}\colon\pre(\gaunt_{n}^{\omega})\to\pre(\mathcal{R})$ the precomposition with the functor $i$. Consider $\mathcal{C}\mathrel{\mathop{:}}=U^{-1}\pre(\mathcal{R})$ along with the restriction $f$ of $i^{\ast}$ to the representable objects; then the pair $(\mathcal{C},f)$ satisfies Axiom (C.3) if and only if $K$ enjoys the following condition.