# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example 5.3.2.10 (Disjoint Unions). Let $I$ be a set, which we regard as a category having only identity morphisms. Let $\mathscr {F}: I \rightarrow \operatorname{Set_{\Delta }}$ be a functor, which we identify with a collection of simplicial sets $\{ X_ i \} _{i \in I}$. Then the comparison map

$\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} ) = \coprod _{i \in I} X_ i$

is an isomorphism of simplicial sets.