Definition 7.7.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathscr {F}, \mathscr {G}: K \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by the same simplicial set $K$. We say that a natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ is cartesian if, for every edge $e: x \rightarrow y$ of the simplicial set $K$, the corresponding diagram
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\begin{equation} \begin{gathered}\label{equation:cartesian-natural-transformation} \xymatrix { \mathscr {F}(x) \ar [r]^{ \mathscr {F}(e) } \ar [d]^{ \gamma _ x } & \mathscr {F}(y) \ar [d]^{ \gamma _{y} } \\ \mathscr {G}(x) \ar [r]^{ \mathscr {F}(e) } & \mathscr {G}(y) } \end{gathered} \end{equation}
is a pullback square in the $\infty $-category $\operatorname{\mathcal{C}}$.