ScalingStacks

Proof. We denote the $\infty$-category $\mathcal{D}\times_{\mathcal{C}}\mathcal{C}_{X/}$ by $\mathcal{D}_{X/}$. Let $\eta\colon X\to RL\left(X\right)$ be the $X$-component of the unit of the adjunction $L\dashv R$. By T.2.1.2.1, the projections $p_{0}\colon\mathcal{C}_{\eta/}\to\mathcal{C}_{X/}$ and $p_{1}\colon\mathcal{C}_{\eta/}\to\mathcal{C}_{RL\left(X\right)/}$ are left fibrations. Moreover, since $\Delta^{\left\{1\right\}}\hookrightarrow\Delta^{1}$ is right anodyne, the map $p_{1}$ is an equivalence of $\infty$-categories. By T.2.2.3.3, we can choose an inverse $p_{1}^{-1}\colon\mathcal{C}_{RL\left(X\right)/}\to\mathcal{C}_{\eta/}$ to $p_{1}$ that strictly commutes with the projections to $\mathcal{C}$. We obtain a commutative diagram of simplicial sets

There is an induced map from the upper left corner to the pullback of the outer rectangle without the upper left corner, which is another commutative diagram of simplicial sets

Since left fibrations are closed under base change (T.2.1.2.1), the vertical maps are left fibrations over $\mathcal{D}$. Hence, to show that the top map is an equivalence it is enough to show that the induced map on fibers is a homotopy equivalence (T.2.2.3.3). For every $Y\in\mathcal{D}$ we get a map

$$\operatorname{Map}_{\mathcal{D}}^{R}\left(L\left(X\right),Y\right)\to\operatorname{Map}_{\mathcal{C}}^{R}\left(X,R\left(Y\right)\right),$$

which is by construction obtained by applying the functor $R$ and pre-composing with the unit $\eta\colon X\to RL\left(X\right)$. By the universal property of the unit map this is a homotopy equivalence for all $Y\in\mathcal{D}$ and therefore the map $\mathcal{D}_{L\left(X\right)/}\to\mathcal{D}_{X/}$ is an equivalence of $\infty$-categories. ∎