ScalingStacks

Remark 3.6. The previous lemma now implies that the category $\gaunt_{n}$ can be identified with the category of Ind-objects of the full subcategory $\gaunt_{n}^{\omega}\subset\gaunt_{n}$ spanned by the compact objects of $\gaunt_{n}$. That is [31, Corollary 2.1.9^′], for any category $\mathcal{D}$ that admits all filtered colimits, if $\Fun^{\omega}(\gaunt_{n},\mathcal{D})$ denotes the full subcategory of $\Fun(\gaunt_{n},\mathcal{D})$ spanned by those functors that preserve filtered colimits, then the restriction functor

$$\Fun^{\omega}(\gaunt_{n},\mathcal{D})\to\Fun(\gaunt_{n}^{\omega},\mathcal{D})$$

is an equivalence.