Kerodon

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Definition 4.4.4.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be a simplicial set, and suppose we are given a pair of diagrams $f,f': X \rightarrow \operatorname{\mathcal{C}}$. A natural transformation from $f$ to $f'$ is a morphism $u: f \rightarrow f'$ in the $\infty $-category $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$. A natural isomorphism from $f$ to $f'$ is a natural transformation $u: f \rightarrow f'$ which is an isomorphism in the $\infty $-category $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$ (Definition 1.3.6.1). We say that $f$ and $f'$ are naturally isomorphic if there exists a natural isomorphism from $f$ to $f'$.