Corollary 9.3.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, and let $f: X \rightarrow Y$ be an $n$-truncated morphism of $\operatorname{\mathcal{C}}$. Suppose we are given a diagram $F: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}_{/X}$. If the simplicial set $K$ is $(n+1)$-connective, then $F$ is a limit diagram in $\operatorname{\mathcal{C}}_{/X}$ if and only if the composition
\[ K^{\triangleleft } \rightarrow { F} \operatorname{\mathcal{C}}_{/X} \rightarrow { f_!} \operatorname{\mathcal{C}}_{/Y} \]
is a limit diagram, where $f_{!}$ is the functor given by composition with $f$ (see Example 4.3.6.15). In particular, the functor $f_{!}$ preserves limits indexed by $(n+1)$-connective simplicial sets.