Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 1.4.2.2. In the case where $K_{\bullet }$ is an $\infty $-category, Definition 1.4.2.1 is superfluous: a $K_{\bullet }$-indexed diagram in $\operatorname{\mathcal{C}}$ (in the sense of Definition 1.4.2.1) is just a functor from $K_{\bullet }$ to $\operatorname{\mathcal{C}}$ (in the sense of Definition 1.4.0.1). However, the redundant terminology will be useful to signal a shift in emphasis. We will generally refer to a map $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ as a functor when we wish to regard the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ on an equal footing. By contrast, we will refer to a map $f: K_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ as a diagram if we are primarily interested in the $\infty $-category $\operatorname{\mathcal{C}}$ (in many cases, the source of $f$ will be a very simple simplicial set).