= List-monad algebras are monoids
{title2=$\mathbf{Set}^{\mathrm{List}}\cong\mathbf{Mon}$}
An <algebra for a monad> for the <list monad> is precisely a <monoid>. For an action $a$, the identity is $a([])$ and multiplication is $a([x,y])$. The algebra laws force the unit and associativity laws and determine $a$ as ordered multiplication. Conversely ordered multiplication defines a list action. Algebra morphisms are exactly <monoid homomorphisms>, giving an isomorphism of categories $\mathbf{Set}^{\mathrm{List}}\cong\mathbf{Mon}$.
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