[[Category theory MOC]]
# Free category

**Free categories** are the [[Free-forgetful adjunction|free objects]] in [[Category of small categories]], #m/def/cat 
forming the [[Adjoint functor|left adjoint]] to the forgetful functor $U : \Cat \to \cat{Quiv}$ to the [[Underlying quiver]]
$$
\begin{align*}
C \dashv U : \Cat \to \cat{Quiv}
\end{align*}
$$
The free category $C\Gamma = \underline{\Gamma}$ is constructed by considering all _composable_ words, called **paths**, as morphisms.

## Universal property

If $\cat D$ is a [[Small category]] with [[Underlying quiver]] $U\cat D$
and $f \in \cat{Quiv}(\Gamma,U\cat D)$ is a [[quiver homomorphism]] then there exists a unique adjunct $g \in \Cat(C\Gamma,\cat D)$ such that the following diagram commutes:

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