Definition 1.5.2.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A diagram in $\operatorname{\mathcal{C}}$ is a map of simplicial sets $f: K \rightarrow \operatorname{\mathcal{C}}$. We will also refer to a map $f: K \rightarrow \operatorname{\mathcal{C}}$ as a diagram in $\operatorname{\mathcal{C}}$ indexed by $K$, or a $K$-indexed diagram in $\operatorname{\mathcal{C}}$.
If $\operatorname{\mathcal{C}}$ is an ordinary category, then a ($K$-indexed) diagram in $\operatorname{\mathcal{C}}$ is a ($K$-indexed) diagram in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
In the special case where $K$ is the nerve $\operatorname{N}_{\bullet }(I)$ of a partially ordered set $I$ (Remark 1.3.1.10), we will refer to a map $f: K \rightarrow \operatorname{\mathcal{C}}$ as a diagram in $\operatorname{\mathcal{C}}$ indexed by $I$, or an $I$-indexed diagram in $\operatorname{\mathcal{C}}$.