Lemma 9.2 (0MJI) · ScalingStacks

[0MJI]

Lemma 9.2. Suppose $\mathcal{D}$ a small quasicategory, and suppose $\mathcal{C}$ a locally small quasicategory that admits small colimits. Suppose $g\colon\mathcal{D}\to\mathcal{C}$ a dense functor. For any quasicategory $\mathcal{E}$ admitting all small colimits, and let $\Fun^{\mathrm{L}}(\mathcal{C},\mathcal{E})\subset\Fun(\mathcal{C},\mathcal{E})$ denote the full sub-quasicategory consisting of those functors that preserve small colimits. Then the functor $\Fun(\mathcal{C},\mathcal{E})\to\Fun(\mathcal{D},\mathcal{E})$ induced by $g$ restricts to a fully faithful functor

\Fun^{\mathrm{L}}(\mathcal{C},\mathcal{E})\to\Fun(\mathcal{D},\mathcal{E}).

[0MJJ]

Proof. The $\infty$ -category $\mathcal{C}$ is a localization of $\pre(\mathcal{D})$ , whence we obtain a fully faithful embedding $\Fun^{\mathrm{L}}(\mathcal{C},\mathcal{E})\to\Fun^{\mathrm{L}}(\pre(\mathcal{D}),\mathcal{E})$ . Now by [28, 5.1.5.6], left Kan extension induces an equivalence $\Fun(\mathcal{D},\mathcal{E})\simeq\Fun^{\mathrm{L}}(\pre(\mathcal{D}),\mathcal{E})$ . See [28, 5.5.4.20]. ∎