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There is a forgetful functor that takes a category to its underlying graph. There is then an adjoint to this that takes the graph to its free category. Can we then take a quotient of this free category to rebuild the original category? This process seems rather important, is it not a familiar set of functors that have special relationships amongst them?

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Denote $F$ for the free category functor and $U$ for the underlying graph functor. Let $\mathcal C$ be a small category. Then the morphisms of $FU(\mathcal C)$ are finite sequences of composable morphisms in $\mathcal C$.

One can put an equivalence relation $\sim$ on the morphisms of $FU(\mathcal C)$: it identifies $(f_1,\dots,f_n)$ and $(g_1,\dots,g_k)$ whenever $$f_n\circ\dots\circ f_1 = g_k \circ \dots \circ g_1 .$$ This equivalence relation clearly is compatible with the composition in $FU(\mathcal C)$, so one can form the quotient $FU(\mathcal C) / {\sim}$ which is isomorphic to $\mathcal C$. The quotient morphism $$ FU(\mathcal C) \to \mathcal C $$ is nothing less than the (component at $\mathcal C$ of the) counit of the adjoint pair $(F,U)$.

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