Definition 7.1.1.17 (Limit and Colimit Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every simplicial set $K$, precomposition with the projection map $K \rightarrow \Delta ^0$ determines a functor
We will refer to $\delta $ as the diagonal functor: it carries each object $X \in \operatorname{\mathcal{C}}$ to the constant diagram $\underline{X}: K \rightarrow \operatorname{\mathcal{C}}$ taking the value $X$.
We say that a functor $T: \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ is a colimit functor for $\operatorname{\mathcal{C}}$ if it is left adjoint to $\delta $, and that $T$ is a limit functor for $\operatorname{\mathcal{C}}$ if it is right adjoint to $\delta $. Note that either of these conditions characterizes the functor $T$ up to isomorphism (Remark 6.2.1.19).