Remark 4.3.1.12. Let $\operatorname{\mathcal{C}}$ be a category and let $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a diagram in $\operatorname{\mathcal{C}}$. If $F$ admits a limit $S = \varprojlim _{I \in \operatorname{\mathcal{K}}} F(I)$, then the slice category $\operatorname{\mathcal{C}}_{/F}$ is isomorphic to $\operatorname{\mathcal{C}}_{/S}$. Similarly, if $F$ admits a colimit $S' = \varinjlim _{I \in \operatorname{\mathcal{K}}} F(I)$, then the coslice category $\operatorname{\mathcal{C}}_{F/}$ is isomorphic to $\operatorname{\mathcal{C}}_{S/}$. In ยง7.1, we will use this observation to extend the theory of limits and colimits to the setting of $\infty $-categories.
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