**Question:**

Let $X$ be a topological space, $U_*\rightarrow X$ be an augmented simplicial space, and let $M_n(U_*)$ be the n-th matching object computed in $sTop$ while $M_n^X(U_*)$ denotes the $n$-th matching object computed in the category $s(Top \downarrow X)$ of simplicial spaces over $X$.

How does one see that for $n \geq 2$, $M_n^X(U_*)$ is isomorphic to $M_n(U_*)$?

(This is claimed in the paragraph following Def. 4.1 of [DI] but is not obvious to me.)

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**Background definitions:**

For a category $C$, let $sC$ denote the category of simplicial objects in $C$ and and $s_{\leq n}C$ denote the category of truncated simplicial objects in $C$ of dimension $n$.

Let $$sk_n: sC \rightarrow s_{\leq n}C$$ be the forgetful functor and let $cosk_n : s_{\leq n}C \rightarrow sC$ denote its right adjoint.

Finally, for $U_* \in sC$ , define the $\textbf{n-th matching object, } M_n(U_*),$ to be the $n$-th object of $cosk_{n-1}(sk_{n-1}(U_*))$.

**Reference:**

[DI] Dugger-Isaksen "Hypercovers in topology"

(https://arxiv.org/abs/math/0111287)