ScalingStacks

Proof. Let $p\colon\mathcal{M}\to\Delta^{1}$ be the coCartesian fibration associated with the functor $L$ (which is also Cartesian, since $L$ has a right adjoint). We can assume that we have a commutative diagram

such that $s|_{\Delta^{\left\{0\right\}}\times\mathcal{C}}=\operatorname{Id}$, $s|_{\Delta^{\left\{1\right\}}\times\mathcal{C}}=L$ and $s|_{\Delta^{1}\times\left\{X\right\}}$ is a coCartesian edge of $\mathcal{M}$ for every $X\in\mathcal{C}$ (combine T.5.2.1.1 and T.5.2.1.3). It is clear from T.1.2.9.2 that for any pair of $\infty$-categories with objects $X\in\mathcal{C}$ and $Y\in\mathcal{D}$ there is a canonical isomorphism

$$\left(\mathcal{C}\times\mathcal{D}\right)_{\left(X,Y\right)/}\simeq\mathcal{C}_{X/}\times\mathcal{D}_{Y/}.$$

Hence, we get an induced commutative diagram

The functor $p_{X}$ is a Cartesian and coCartesian fibration by the duals of T.2.4.3.1(1) and T.2.4.3.2(1). Moreover, an edge in $\mathcal{M}_{X/}$ is (co)Cartesian if and only if its projection to $\mathcal{M}$ is (co)Cartesian by the duals of T.2.4.3.1(2) and T.2.4.3.2(2), which shows that the functor $L_{X}$ is associated with $p_{X}$. It follows that $L_{X}$ has a right adjoint $R_{X}$.

Assuming that $R$ is fully faithful, we will show that $R_{X}$ is fully faithful by showing that the counit of the adjunction $L_{X}\dashv R_{X}$ is an equivalence. For every object, the counit map is an edge of $\mathcal{M}_{X/}$. Since the projection $\mathcal{M}_{X/}\to\mathcal{M}$ is conservative, it is enough to show that the counit map of $L_{X}\dashv R_{X}$ is mapped to the counit map of $L\dashv R$. Indeed, for an object $Y\in\mathcal{D}\simeq\mathcal{M}|_{\Delta^{\left\{1\right\}}}$, we choose a Cartesian edge $e\colon R\left(Y\right)\to Y$ and a coCartesian edge $d\colon R\left(Y\right)\to L\left(R\left(Y\right)\right)$, and combine them into a commutative diagram of the form:

where $f|_{\Delta^{\left\{0,1\right\}}}=d$ and $f|_{\Delta^{\left\{0,2\right\}}}=e$. Since $d$ is coCartesian, there exists a lift $\overline{f}\colon\Delta^{2}\to\mathcal{M}$ that gives an edge

$$\overline{f}|_{\Delta^{\left\{1,2\right\}}}=c\colon L\left(R\left(Y\right)\right)\to Y$$

that is isomorphic to the counit map of the adjunction $L\dashv R$ at $Y$ in the homotopy category $h\mathcal{D}$. We can similarly construct the counit map for an object of $\mathcal{M}_{X/}$. The assertion now follows from the above characterization of (co)Cartesian edges in $\mathcal{M}_{X/}$. ∎