ScalingStacks

Remark 4.1.9.

Let \(\Pr^L_{\mathbb V}\) denote the (non-full) subcategory of \(\widehat{\mathrm{Cat}}[\mathbb V]\) on the presentably \(\mathbb V\)-enriched \(\infty\)-categories \(\mathcal C\) and on those \(\mathbb V\)-enriched functors that are left adjoint in the \(\mathbb V\)-enriched sense,¹⁷see [MS21, Def. A.2.12]. Then, it is shown in [MS21, Thm. A.3.8] that the functor \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L}) \rightarrow\widehat{\mathrm{Cat}}[\mathbb V]\) factors as an equivalence through \(\Pr^L_{\mathbb V}\). In particular, \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L}) \simeq \Pr^L_{\mathbb V}\) is a subcategory of \(\widehat{\mathrm{Cat}}[\mathbb V]\); it is merely a property of large \(\mathbb V\)-enriched categories and \(\mathbb V\)-enriched functors to be in the image of \(\mathrm{Mod}_{\mathbb V}(\mathrm{Pr}^\mathrm{L}) \rightarrow\widehat{\mathrm{Cat}}[\mathbb V]\).