I'm trying to do an exercise on adjoints, where one of the questions asks to prove that the forgetful functor $U: \mathsf{Magma} \rightarrow \mathsf{Set}$ has a left adjoint.
Is there a way of proving this, assuming that both Magma and Set are categories?
On
Yes. In general, for any algebraic theory $\mathcal A$ (that is magmas, monoids, groups, etc.. ) their is an adjunction $$U:\mathcal A \to \mathcal Set$$ $$F:\mathcal Set\to\mathcal A$$ with right adjoint given by a forgetful functor $U$ and left adjoint given by the free algebra functor $F$. Formally, a free algebra on a set $X$ is defined by the left adjoint as $F(X)$. But intuitively, a free algebra on a set $X$ is smallest algebra containing $X$ which is "freely" generated by the terms of the algebra.
So to get the left adjoint for the theory of magmas, you need to ask yourself what is a free magma generated by a set.
Hint:
A free monoid on a set $X$ is the monoid of words with alphabet $X$. If you remove associativity and the identity, you get trees with $X$-labelled leaves. For example, if $X=\{a,b,c\}$, then the underlying set of the free magma on $X$ is given the set of terms $$U_{\text{Magma}}(F_{\text{Magma}}(X))=\{a,b,c,\mu(a,c),\mu(a,\mu(\mu(a,b),c)),\ldots etc.\}$$
Magmas are algebraic structures, and, as such, enjoy the existence of free objects. Do you see why this implies the existence of a particular left adjoint?