Naturality of the cap product ($f_{\ast }(x⌢f^{\ast }\alpha )=f_{\ast }x⌢\alpha$)

For a singular simplex $\sigma$ of dimension $n$ and a degree-$p$ cochain, use $\sigma ⌢\alpha =\alpha (\sigma |[v_0,\dots ,v_p])\sigma |[v_p,\dots ,v_n]$, 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 $f_{\ast }(\sigma ⌢f^{\ast }\alpha )=(f\circ \sigma )⌢\alpha$ already on chains. Passing to homology and cohomology proves the displayed cap product identity.