[05Y1] Proposition 3.1.8 ([SY19, Theorem 3.12]). The inclusion 𝐎𝐩d↪𝐎𝐩∞\mathbf{Op}_{d}\hookrightarrow\mathbf{Op}_{\infty} admits a left adjoint hdh_{d}, such that for every ∞\infty-operad 𝒪\mathcal{O}, the unit transformation θd:𝒪→hd𝒪\theta_{d}\colon\mathcal{O}\to h_{d}\mathcal{O} is essentially surjective and for all X1,…,Xn,Y∈𝒪¯X_{1},\dots,X_{n},Y\in\underline{\mathcal{O}}, the map of spaces Mul𝒪({X1,…,Xn};Y)→Mulhd𝒪({θd(X1),…,θd(Xn)};θd(Y))\operatorname{Mul}_{\mathcal{O}}\left(\left\{X_{1},\dots,X_{n}\right\};Y\right)\to\operatorname{Mul}_{h_{d}\mathcal{O}}\left(\left\{\theta_{d}\left(X_{1}\right),\dots,\theta_{d}\left(X_{n}\right)\right\};\theta_{d}\left(Y\right)\right) is the (d−1)\left(d-1\right)-truncation map.