ScalingStacks

Remark 4.1.2. Let us denote the horizontal morphisms in the above diagram by $f\colon A\to X$ and $g\colon B\to Y$. By the dual of T.5.5.5.12 we have a homotopy fiber sequence

$$\operatorname{Map}_{\mathcal{C}_{A//\overline{Y}}}\left(B,X\right)\to\operatorname{Map}_{\mathcal{C}_{A/}}\left(B,X\right)\to\operatorname{Map}_{\mathcal{C}_{A/}}\left(B,Y\right)$$

over $g\in\operatorname{Map}_{\mathcal{C}_{A/}}\left(B,Y\right)$. Using T.5.5.5.12 again for the middle and the right term we obtain a presentation of $\operatorname{Map}_{\mathcal{C}_{A//\overline{Y}}}\left(B,X\right)$ as the total fiber of the square

In other words, we have a homotopy fiber sequence

$$L\left(q\right)\to\operatorname{Map}_{\mathcal{C}}\left(B,X\right)\to\operatorname{Map}_{\mathcal{C}}\left(A,X\right)\times_{\operatorname{Map}_{\mathcal{C}}\left(A,Y\right)}^{h}\operatorname{Map}_{\mathcal{C}}\left(B,Y\right)$$

over the point determined by the diagram $q$.

Another reasonable definition of the space of lifts is as follows. The inclusion $\Delta^{\left\{0,1\right\}}\times\Delta^{\left\{0,2\right\}}\hookrightarrow\Delta^{3}$ induces a restriction map $\mathcal{C}^{\Delta^{3}}\to\mathcal{C}^{\Delta^{1}\times\Delta^{1}}$ and we can consider the (automatically homotopy) fiber over the vertex $q\in\mathcal{C}^{\Delta^{1}\times\Delta^{1}}$, which is an $\infty$-category. In T.5.2.8.22 it is proved that this $\infty$-category is categorically equivalent to $L\left(q\right)$ (and in particular a Kan complex).