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The fully faithful functor from the category of monoids to the category of categories admits a left adjoint. Given a category , we can construct a monoid which is freely generated by the set of morphisms of , modulo the relations for each object and for each pair and of composable morphisms in .
Is there a reference for this fact? I remember learning it from a mathstackexchange/mathoverflow question, but I cannot seem to locate it again. It very possible there is also a standard textbook exercise or lemma which proves it.
I don't have a reference for you. Instead, let me explain how it comes from very general considerations about essentially algebraic theories. I'm hoping this is interesting to you nevertheless.
Let me write and for the essentially algebraic theories of monoids and categories respectively. These are the opposite of categories of finitely presented monoids and finitely presented categories. We see a monoid as a finite limit-preserving functor
I define a category internal to
and the rest of the data is determined. This yields a finite limit-preserving functor
The precomposition functor takes a monoid and produces a category
Therefore, is the functor that you are interested in. Now, for any such finite limit-preserving , the precomposition functor has a left adjoint, which is the left Kan extension along -- indeed, it preserves the preservation of finite limits. This is the left adjoint you mentioned!