[[Category theory MOC]]
# Underlying quiver

Let $\cat C$ be a [[Small category]]. The **underlying quiver** $U(\cat C)$ is the [[quiver]] whose vertices are $\Ob(\cat C)$ and whose edges are the morphisms of $\cat C$, #m/def/cat
such that if $f \in \cat C(X,Y) \iff f \in U(\cat C)(X,Y)$.
Its [[Adjoint functor|left-adjoint]] is the [[Free category]].

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