A causal-net is a finite acyclic directed graph. In this paper, we introduce a category, denoted as $\mathbf{Cau}$, whose objects are causal-nets and morphisms are functors of path categories of causal-nets. It is called causal-net category and in fact the Kleisli category of the "free category on a causal-net" monad. We study several composition-closed classes of morphisms in $\mathbf{Cau}$, which characterize interesting causal-net relations, such as coarse-graining, immersion-minor,

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... al minor, etc., and prove several useful decomposition theorems. In addition, we show that the minor relation can be understood as a special kind of sub-quotients in $\mathbf{Cau}$. Base on these results, we conclude that $\mathbf{Cau}$ is a natural setting for studying causal-nets, and the theory of $\mathbf{Cau}$ should shed new light on the category-theoretic understanding of graph theory.