[0MJJ] Proof. The β\infty-category π\mathcal{C} is a localization of π«β‘(π)\pre(\mathcal{D}), whence we obtain a fully faithful embedding FunLβ‘(π,β°)βFunLβ‘(π«β‘(π),β°)\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 Funβ‘(π,β°)βFunLβ‘(π«β‘(π),β°)\Fun(\mathcal{D},\mathcal{E})\simeq\Fun^{\mathrm{L}}(\pre(\mathcal{D}),\mathcal{E}). See [28, 5.5.4.20]. β