# Kerodon

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Remark 5.2.6.9. Let $n \geq 1$ and let $\overrightarrow {X}: [n] \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets which we denote by

$X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n).$

Let $\overrightarrow {X}': [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the constant diagram taking the value $X(0)$. Let $\overrightarrow {X}_0 \subseteq \overrightarrow {X}$ be the subfunctor given by the diagram

$\emptyset \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n),$

and define $\overrightarrow {X}'_0 \subseteq \overrightarrow {X}'$ similarly, so that we have a pushout diagram

$\xymatrix@R =50pt@C=50pt{ \overrightarrow {X}'_0 \ar [r] \ar [d] & \overrightarrow {X}_0 \ar [d] \\ \overrightarrow {X}' \ar [r] & \overrightarrow {X} }$

in the category $\operatorname{Fun}( [n], \operatorname{Set_{\Delta }})$. Combining Remark 5.2.6.8, Example 5.2.6.7, and Example 5.2.6.6, we obtain a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ) \times X(0) \ar [r] \ar [d] & M( X(1) \rightarrow \cdots \rightarrow X(n) ) \ar [d] \\ \Delta ^ n \times X(0) \ar [r] & M( X(0) \rightarrow \cdots \rightarrow X(n) ). }$