Multiplicativity of mapping degree ($\mathrm{deg}(g\circ f)=\mathrm{deg}g\mathrm{deg}f$)

For maps of oriented connected closed manifolds of the same dimension, apply functoriality of homology to $f_{\ast }[M]=(\mathrm{deg}f)[N]$ and $g_{\ast }[N]=(\mathrm{deg}g)[L]$. The composite coefficient is their product. This recovers the multiplication of signed sheet counts for compositions of finite covering maps.