A very standard example of left adjoints would be the functor $F:Sets\to Grps$ that maps sets to their free groups is a left adjoint to $G:Grps \to Sets$ which forgets the group structure. My question is how can we construct this left adjoint? When I saw the construction of the free group, I didn't really understand what was happening. But based on this construction, I am assuming that in order to construct a left adjoint (without knowing what it is), we need to prove the existence? But if we show that it exists, then how do we necessarily know that it is $F$ from above. I am just confused on the whole idea of constructions from universal properties. Any help would be appreciated. I am very new to the subject so please try to make things as elementary as possible.
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2$F$ sends a set to the free group generated by the elements of the set. The construction of the free group is what you need to understand. Take a look at this post. – John Douma Sep 24 '23 at 18:01