$0\leq k\leq n$, $$\label{eq:Near-sel-v8:1}
    h\big(tv_k+(1-t)z\big)=H_{k}(h(z),t),\quad \text{$z\in
      |\tau_{k+1}|$ and $t\in [0,1]$.}$$ This $h$ is as required. Indeed, take any continuous map $g:|\sigma|\to Y$ satisfying \[item:Near-sel-v8:1\] and . Then $g(v_{n+1})=h(v_{n+1})$ and by , we get that ${g\operatorname{\upharpoonright}|\tau_n|=h\operatorname{\upharpoonright}|\tau_n|}$. Inductively, this implies that $g=g\operatorname{\upharpoonright}|\tau_0|=h\operatorname{\upharpoonright}|\tau_0|=h$.

To see finally that \[item:Near-sel-v8:2\] is equivalent to , take a face $\tau\subset \sigma$ and a vertex $v\in \sigma$ with $v<u$, for every $u\in \tau$. Then $v=v_k$ for some $0\leq k\leq n$, and $\tau\subset \tau_{k+1}$. If $z\in |\tau|\subset |\tau_{k+1}|$ and $t\in [0,1]$, it follows from that $$h\big(tv+(1-t)z\big)=h\big(tv_k+(1-t)z\big)=
    H_k(h(z),t)=H_v(h(z),t).\qedhere$$

Let $\Sigma$ be an oriented simplicial complex with respect to some partial order $\leq$ on $\Sigma^0$, and $\Phi,\Psi:\Sigma^0\operatorname{\leadsto}Y$ be as in Theorem \[theorem-Near-sel-v7:1\]. For each vertex ${v\in \Sigma^0}$, fix a contraction $H_v:\Phi(v)\times[0,1]\to \Psi(v)$ into some point ${p_v\in \Psi(v)}$. Since each simplex $\sigma\in \Sigma$ is oriented with respect to $\leq$, by Proposition \[proposition-Near-sel-v8:1\] applied with $\Phi\operatorname{\upharpoonright}\sigma$, $\Psi\operatorname{\upharpoonright}\sigma$ and the fixed contractions $H_v$, $v\in \sigma$, there exists a unique continuous map $h_\sigma:|\sigma|\to Y$ satisfying both properties \[item:Near-sel-v8:1\] and \[item:Near-sel-v8:2\] in that proposition. Suppose that $\sigma_1,\sigma_2\in \Sigma$ are simplices which have a common face $\tau=\sigma_1\cap
  \sigma_2$. Then the linear order $\leq$ on $\tau$ is the same in each one of the oriented simplices $(\sigma_i,\leq)$, $i=1,2$. 