$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.7.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{\gamma }: \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ be a natural transformation between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The natural transformation $\overline{\gamma }$ is cartesian.
- $(2)$
The diagram (7.80) is a levelwise pullback square.
Proof.
For each vertex $x \in K$, let $e_{x}$ denote the edge of $K^{\triangleright }$ given by the map $\Delta ^1 \simeq \{ x\} ^{\triangleright } \hookrightarrow K^{\triangleright }$. Note that $\overline{\gamma }$ carries $e_ x$ to a commutative diagram
\[ \xymatrix { \mathscr {F}(x) \ar [r] \ar [d]^{\gamma _{x}} & C \ar [d]^{f} \\ \mathscr {G}(x) \ar [r] & D } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, which is obtained from (7.80) by evaluating at $x$. It follows that $(1)$ implies $(2)$. Conversely, suppose that condition $(2)$ is satisfied, and let $e: x \rightarrow y$ be any edge of $K^{\triangleright }$; we wish to show that $\overline{\gamma }$ carries $e$ to a pullback square in $\operatorname{\mathcal{C}}$. Without loss of generality, we may assume that $e$ is nondegenerate (Remark 7.7.1.3). If $y$ is the cone point of $K^{\triangleright }$, then $x$ belongs to $K$ and $e = e_ x$, so the desired result follows from assumption $(2)$. We are therefore reduced to the case where $e$ is an edge of $K$: that is, to showing that the natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ is cartesian. This follows from Remark 7.7.1.11, since $\gamma $ is a pullback of the cartesian natural transformation $\underline{f}: \underline{C} \rightarrow \underline{D}$ (Example 7.7.1.6).
$\square$