= Naturality of the cap product
{title2=$f_*(x\frown f^*\alpha)=f_*x\frown\alpha$}
For a <singular simplex> $\sigma$ of dimension $n$ and a degree-$p$ <cochain>, use $\sigma\frown\alpha=\alpha(\sigma|[v_0,\ldots,v_p])\sigma|[v_p,\ldots,v_n]$, setting the result to zero if $p>n$. A <continuous map> $f$ commutes with both face restrictions, and the <pullback> cochain evaluates by composing with $f$. Therefore $f_*(\sigma\frown f^*\alpha)=(f\circ\sigma)\frown\alpha$ already on chains. Passing to <homology> and <cohomology> proves the displayed <cap product> identity.
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