Remark 1.5.2.2. In the case where $K$ is an $\infty $-category, Definition 1.5.2.1 is superfluous: a $K$-indexed diagram in $\operatorname{\mathcal{C}}$ (in the sense of Definition 1.5.2.1) is just a functor from $K$ to $\operatorname{\mathcal{C}}$ (in the sense of Definition 1.5.0.1). 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 morphism $f: K \rightarrow \operatorname{\mathcal{C}}$ as a *diagram* if we are primarily interested in the $\infty $-category $\operatorname{\mathcal{C}}$ (in many cases, $K$ will be a very simple simplicial set).

$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$