ScalingStacks

Proof. For ease of notation, set $\mathcal{D}=\mathcal{C}_{A/}$. Recall that

$$L\left(q\right)=\operatorname{Map}_{\mathcal{D}_{/\overline{Y}}}\left(\overline{B},\overline{X}\right)$$

and therefore

$$L\left(q\right)\times L\left(q\right)=\operatorname{Map}_{\mathcal{D}_{/\overline{Y}}}\left(\overline{B},\overline{X}\right)\times\operatorname{Map}_{\mathcal{D}_{/\overline{Y}}}\left(\overline{B},\overline{X}\right)\simeq\operatorname{Map}_{\mathcal{D}_{/\overline{Y}}}\left(\overline{B},\overline{X}\times\overline{X}\right).$$

Products in the over-category are fibered products and products in the under-category are just ordinary products (dual of T.1.2.13.8). Hence, $\overline{X}\times\overline{X}$ is the diagram $A\to X\times_{Y}X\to Y$, which we denote by $\overline{X\times_{Y}X}$. Thus, a point $s=\left(s_{0},s_{1}\right)\in L\left(q\right)\times L\left(q\right)$ corresponds to a lift in the diagram

in the category $\mathcal{D}$. Furthermore, the diagonal map $\delta_{L\left(q\right)}\colon L\left(q\right)\to L\left(q\right)\times L\left(q\right)$ is induced from the diagonal map $\delta_{\overline{X}}\colon\overline{X}\to\overline{X}\times\overline{X}$. Namely, $\delta_{L\left(q\right)}=\left(\delta_{\overline{X}}\right)_{*}$. Our goal is therefore to compute the homotopy fiber of $\left(\delta_{\overline{X}}\right)_{*}$ over a given point

$$s=\left(s_{0},s_{1}\right)\simeq\operatorname{Map}_{\mathcal{D}_{/\overline{Y}}}\left(\overline{B},\overline{X}\times\overline{X}\right).$$

The projection $\mathcal{D}_{/\overline{Y}}\to\mathcal{D}$ induces an equivalence

$$\left(\mathcal{D}_{/\overline{Y}}\right)_{/\overline{X\times_{Y}X}}\simeq\mathcal{D}_{/\overline{X\times_{Y}X}}.$$

It follows that the fiber is the space of lifts in the diagram

in $\mathcal{D}$. By (the dual of) T.5.5.5.12, this space of lifts is homotopy equivalent to the mapping space $\operatorname{Map}_{\mathcal{D}_{/\overline{X\times_{Y}X}}}\left(\overline{B},\overline{X}\right)$. Recalling that $\mathcal{D}=\mathcal{C}_{A/}$, we see that this is none other than the space of lifts for $p$. ∎