Kerodon

Remark 4.4.4.2. In the situation of Definition 4.4.4.1, a natural transformation from $f$ to $f'$ is simply a homotopy from $f$ to $f'$, in the sense of Definition 3.1.5.2: that is, a map of simplicial sets $h: \Delta ^1 \times X \rightarrow \operatorname{\mathcal{C}}$ satisfying $h|_{ \{ 0\} \times X} = f$ and $h|_{ \{ 1\} \times X } = f'$. However, the terminology of Definition 4.4.4.1 is intended to signal a shift in emphasis. We will generally reserve use of the term homotopy between diagrams $f,f': X \rightarrow \operatorname{\mathcal{C}}$ for the case where $\operatorname{\mathcal{C}}$ is a Kan complex, and use the term natural transformation when $\operatorname{\mathcal{C}}$ is a more general $\infty $-category.