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Find a left adjoint to the forgetful functor from the category of monoids to the category of sets.

I am not sure what to do here. Clearly we must have something akin to free product... I'm sorry I cannot say more, and any help would be greatly appreciated.

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  • $\begingroup$ You are on the right track. So, define a functor $F$ from Set to Mon by sending a set $S$ to the free monoid generated by $S$. Now, there will only be one way to define $F$ on arrows, such that the UMP of the adjoint holds. $\endgroup$ – Matematleta Oct 22 '18 at 4:42
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The free monoid on a set $X$ is the collection of finite words with letters from $X$. We can denote it as $X^*$. Multiplication is concatenation, and the identity is the empty word.

Each set map $X\to M$, where $M$ is a monoid, extends uniquely to a monoid map $X^*\to M$, so we have a natural isomorphism $$\textrm{Map}_{\textrm{Monoid}}(X^*,M)\cong \textrm{Map}_{\textrm{Set}}(X,M)$$ confirming adjointness.

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