Homomorphism commutes with word maps

Statement

Suppose $w$ is a word in the letters $x_1,x_2,\dots ,x_n$ (these are just formal symbols). Suppose $\phi :G\to H$ is a homomorphism of groups. Then, $\phi$ commutes with $w$, i.e.:

$\phi (w(g_1,g_2,\dots ,g_n))=w(\phi (g_1),\phi (g_2),\dots ,\phi (g_n))\mathrm{\forall }{g}_1,g_2,\dots ,g_n\in G$

where the $w$ on the left is the word map in $G$ (i.e., it evaluates the word for a tuple of values of the letters in $G$ and the $w$ on the right is the word map in $H$.

Proof

A formal proof can be given by inducting on the length of the word. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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