Proposition 7.1.2.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{\rho }: \overline{F} \rightarrow \overline{G}$ be a natural transformation between diagrams $\overline{F}, \overline{G}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Assume that, for every vertex $x \in K$, the morphism $\overline{\rho }_{x}: \overline{F}(x) \rightarrow \overline{G}(x)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then any two of the following conditions imply the third:
- $(1)$
The morphism of simplicial sets $\overline{F}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(2)$
The morphism of simplicial sets $\overline{G}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(3)$
The natural transformation $\overline{\rho }$ carries the cone point ${\bf 0} \in K^{\triangleleft }$ to an isomorphism $\overline{\rho }_{\bf 0}: \overline{F}( {\bf 0} ) \rightarrow \overline{G}( {\bf 0} )$.
Proof.
Set $F = \overline{F}|_{K}$ and $G = \overline{G}|_{K}$, so that $\overline{\rho }$ restricts to an isomorphism $\rho : F \rightarrow G$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ (Theorem 4.4.4.4). Set $X = \overline{F}( {\bf 0} )$ and $Y = \overline{F}( {\bf 0} )$, and let $\underline{X}, \underline{Y}: K \rightarrow \operatorname{\mathcal{C}}$ be the constant maps taking the values $X$ and $Y$, respectively. Let $c$ denote the composition $\Delta ^1 \times K \simeq K \star _{K} K \rightarrow \Delta ^{0} \star _{ \Delta ^{0} } K = K^{\triangleleft }$. Then the composition
\[ \Delta ^1 \times \Delta ^1 \times K \xrightarrow { \operatorname{id}\times c} \Delta ^1 \times K^{\triangleleft } \xrightarrow { \overline{\rho } } \operatorname{\mathcal{C}} \]
can be identified with a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ \underline{X} \ar [d]^{ \alpha } \ar [dr]^{\gamma } \ar [r]^-{ \underline{f} } & \underline{Y} \ar [d]^{\beta } \\ F \ar [r]^-{ \rho }_{\sim } & G } \]
in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Using Remark 7.1.2.6, we can reformulate conditions $(1)$ and $(2)$ as follows:
- $(1')$
The natural transformation $\alpha $ exhibits $X$ as a limit of $F$.
- $(2')$
The natural transformation $\beta $ exhibits $Y$ as a limit of $G$.
Since $\rho $ is an isomorphism, we can use Remark 7.1.1.8 to reformulate $(1')$ as follows:
- $(1'')$
The natural transformation $\gamma $ exhibits $X$ as a limit of $G$.
It will therefore suffice to show that any two of the conditions $(1'')$, $(2')$, and $(3)$ imply the third, which is a special case of Remark 7.1.1.9.
$\square$