Variant 7.7.1.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and suppose we are given a natural transformation $\overline{\gamma }: \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. If there exists a morphism $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ and a levelwise pullback square $\sigma :$
in $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$, then $\overline{\gamma }$ is cartesian (see Remark 7.7.1.11 and Example 7.7.1.6). Conversely, suppose that $\overline{\gamma }$ is cartesian. Let $v$ denote the cone point of $K^{\triangleright }$ and set $C = \overline{\mathscr {F}}(v)$ and $D = \overline{\mathscr {G}}(v)$, so that $\overline{\gamma }$ determines a morphism $f: C \rightarrow D$. There is a unique morphism of simplicial sets $h: \Delta ^1 \times K^{\triangleright } \rightarrow K^{\triangleright }$ such that $h|_{ \{ 0\} \times K^{\triangleright } }$ is the identity and $h|_{ \{ 1\} \times K^{\triangleright } }$ is the constant map taking the value $v$. Precomposition with $h$ carries $\overline{\gamma }$ to a morphism in the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times K^{\triangleright }, \operatorname{\mathcal{C}})$, which we can identify with a levelwise pullback square of the form (7.81).