ScalingStacks

Proof. Recall the combinatorial simplicial model category $\mathbf{POp}_{\infty}$ of $\infty$-preoperads, whose underlying $\infty$-category is $\mathbf{Op}_{\infty}$ (see A.2.1.4). Let $\overline{\mathcal{Z}}_{0}\subseteq\overline{\mathcal{Z}}_{1}\subseteq\mathbf{Fin}_{*}$ be the following subcategories:

1. (1)

   The category $\overline{\mathcal{Z}}_{0}$ is discrete and contains only the objects $\left\langle 1\right\rangle$ and $\left\langle m\right\rangle$.

2. (2)

   The category $\overline{\mathcal{Z}}_{1}$ contains $\overline{\mathcal{Z}}_{0}$ together with a unique non-identity morphism, which is the active map $\alpha\colon\left\langle m\right\rangle\to\left\langle 1\right\rangle$.

We endow $\overline{\mathcal{Z}}_{0}$ and $\overline{\mathcal{Z}}_{1}$ with the induced (trivial) marking. Unwinding the definition, for any $\infty$-operad $\mathcal{P}$, the simplicial set $\operatorname{Map}_{\mathbf{POp}_{\infty}}\left(\overline{\mathcal{Z}}_{0},\mathcal{P}^{\natural}\right)$ is isomorphic to $\mathcal{P}_{\left\langle m\right\rangle}^{\simeq}\times\mathcal{P}_{\left\langle 1\right\rangle}^{\simeq}$. Moreover, given

$$\underline{X}=\left(X_{1}\oplus\cdots\oplus X_{m},Y\right)\in\mathcal{P}_{\left\langle m\right\rangle}^{\simeq}\times\mathcal{P}_{\left\langle 1\right\rangle}^{\simeq},$$

the fiber of the fibration (hence also the homotopy fiber)

$$\varphi_{\mathcal{P}}\colon\operatorname{Map}_{\mathbf{POp}_{\infty}}\left(\overline{\mathcal{Z}}_{1},\mathcal{P}^{\natural}\right)\to\operatorname{Map}_{\mathbf{POp}_{\infty}}\left(\overline{\mathcal{Z}}_{0},\mathcal{P}^{\natural}\right)\simeq\mathcal{P}_{\left\langle m\right\rangle}^{\simeq}\times\mathcal{P}_{\left\langle 1\right\rangle}^{\simeq}$$

over $\underline{X}$ is homotopy equivalent to the multi-mapping space $\operatorname{Mul}_{\mathcal{P}}\left(\left\{X_{1},\dots,X_{m}\right\};Y\right)$. Let $\mathcal{Z}_{0}$ and $\mathcal{Z}_{1}$ be $\infty$-operads that are fibrant replacements of $\overline{\mathcal{Z}}_{0}$ and $\overline{\mathcal{Z}}_{1}$, respectively. Moreover, let $f\colon\mathcal{Z}_{0}\to\mathcal{Z}_{1}$ be a map corresponding to the inclusion $\overline{\mathcal{Z}}_{0}\hookrightarrow\overline{\mathcal{Z}}_{1}$. The functor $F\colon\left(\mathbf{Op}_{\infty}\right)_{\mathcal{Z}_{0}/}\to\mathcal{S}$, co-represented by $f\colon\mathcal{Z}_{0}\to\mathcal{Z}_{1}$, preserves limits. Furthermore, its value on $g\colon\mathcal{Z}_{0}\to\mathcal{P}$ fits by T.5.5.5.12 into a fiber sequence

which therefore identifies $F\left(\mathcal{P}\right)$ with $\operatorname{Mul}_{\mathcal{P}}\left(\left\{X_{1},\dots,X_{m}\right\};Y\right)$ for the objects $X_{1},\dots,X_{m},Y\in\mathcal{P}$ determined by $g$.

Let $U\colon\mathbf{Op}_{\infty,*}\to\left(\mathbf{Op}_{\infty}\right)_{\mathcal{Z}_{0}/}$ be the functor induced from the map $\mathcal{Z}_{0}\to\mathbf{Triv}$ corresponding to the inclusion $\overline{\mathcal{Z}}_{0}\hookrightarrow\mathbf{Triv}$. By T.1.2.13.8, the functor $U$ preserves limits. We define $G^{(m)}\colon\mathbf{Op}_{\infty,*}\to\mathcal{S}$ to be the composition of $F$ and $U$, which is limit-preserving as a composition of limit preserving-functors. Unwinding the definitions, $G^{(m)}$ indeed lifts $hG^{(m)}$. ∎