Kerodon

Example 7.7.1.2. In the situation of Definition 7.7.1.1, suppose that the natural transformation $\gamma $ is an isomorphism. Then, for every edge $e: x \rightarrow y$ of $K$, the diagram (7.79) is automatically a pullback square, since the vertical maps are isomorphisms (see Corollary 7.6.2.27). It follows that every isomorphism in the $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{C}})$ is a cartesian natural transformation. In particular, for every diagram $\mathscr {F}: K \rightarrow \operatorname{\mathcal{C}}$, the identity transformation $\operatorname{id}: \mathscr {F} \rightarrow \mathscr {F}$ is cartesian.