$H$ is a ${{\mathcal{U}_{}}}$-homotopy.

\[thm:nerve theorem\] Let $\mathcal{K} = \{ K_i \}_{i \in I}$ be a cover of a polyhedron $K$ by subcomplexes. Let $\mathcal{L} = \{ L_i \}_{i \in I}$ be a cover of a polyhedron $L$ by subcomplexes. Let $p \colon K \to L$ be a surjective simplicial map that maps elements of $\mathcal{K}$ into the corresponding elements of $\mathcal{L}$ (i.e. $p(K_i) \subset L_i$ for each $i \in I$). If $\mathcal{K}$ and $\mathcal{L}$ are isomorphic $AE(n)$-covers, then $p$ is a weak $n$-homotopy equivalence.

To begin, we will show that $p$ induces monomorphisms on homotopy groups of dimensions less than $n$. Let $m < n$. Let $\varphi \colon S^m \to K$. Assume that $p \circ \varphi$ is null-homotopic in $L$. Let $\varPhi \colon B^{m+1} \to L$ denote such a null-homotopy ($\varPhi_{\mid S^m} = p \circ \varphi$). We have to show that $\varphi$ is null-homotopic in $K$.

Let ${{\mathcal{F}_{}}}$ be a finite closed cover of $B^{m+1}$ with mesh small enough so that for each $F \in {{\mathcal{F}_{}}}$ we can pick $i_F \in I$ such that $$(\ast)\  \varphi(F \cap S^m) \subset \operatorname{ost}_{\beta K} K_{i_F} \text{ and } 
    (\ast\ast)\  \varPhi(F) \subset \operatorname{ost}_{\beta L} L_{i_F}.$$ By $(\ast\ast)$, a map $C \colon \mathcal{F} \to 