It is well-known that, for a finite quiver $\Gamma$ and its path
algebra $k\Gamma$ over a field $k$, the linear-representation
category {\bf Lin-Rep}$\Gamma$ is equivalent to the
$k\Gamma$-module category $k\Gamma$-Mod . The purpose of this
paper is to
 generalize the conclusion to the so-called set-representation category
{\bf Set-Rep}$\Gamma$ and its equivalent category
$P(\Gamma)$-$\mathcal{SET}^{o}$.

 The authors firstly introduce the definition of the set-representation 
 category
 
{\bf Set-Rep}$\Gamma$ and find out its equivalent category
$P(\Gamma)$-$\mathcal{SET}^{o}$. Secondly, through a  finite
connected quiver $\Gamma$ on which all objects of
$P(\Gamma)$-$\mathcal{SET}^{o}$ are (positively) graded, they find
some interesting relations between the two categories
$k\Gamma$-Mod and $P(\Gamma)$-$\mathcal{SET}^{o}$ (see Corollary
3.8 and Corollary 3.9 ),  although one of them is abelian while
the other is not. Under the equivalence of categories, such
relations also exist between {\bf Lin-Rep}$\Gamma$ and {\bf
Set-Rep}$\Gamma$.