Let $X$ be a topological space, $R$ a ring, $n \in \mathbb{N}$ natural. Let $S_n(X, R) = \bigoplus_{s: \Delta_n \to X} R$ where the $s: \Delta_n \to X$ are the singular n-simplices, therefore continuous maps from standard n-simplex $\Delta_n$ to $X$. Denote futhermore with $S^n(X,R) = Hom_R(S_n(X, R) ,R)$ the dual space to $S_n(X,R)$.
Let $\phi: |K| \to X$ a triangulation of $X$. Choosing total order for $v \in K$ then there exist a map $\alpha$ from the set of n-Simplices $\sigma :=\{x_0, x_1, ... , x_n\}$ to $S_n(X, R)$: we map a n-simplex $\sigma :=\{x_0, x_1, ... , x_n\}$ to singular n-simplex $s: \Delta_n \to X$ defined by $s(t_0, ..., t_n):= \phi(t_0 x_0 +... t_n x_n)$.
My question is why this map induces the dual map $S^n(X, R) \to S^n(|K|, R)$? Indeed, if there would exist a map $S_n(K, R) \to S_n(X, R)$ that would obviously induce such a map because $Hom_R(-, R)$ is a contravariant functor. But why it is induced here by $\alpha$? Can I indentify here the n-simplices of $K$ with free generators of$S_n(|K|,R)$? Therefore interpret all n-simplices of $K$ as continuous maps $\Delta_n \to |K|$?
