  In this paper, we study the componentwise linearity of edge ideals of
weighted oriented graphs. We show that if $D$ is a weighted oriented graph
whose edge ideal $I(D)$ is componentwise linear, then the underlying simple
graph $G$ of $D$ is co-chordal. This is an analogue of Fr\"oberg's theorem for
weighted oriented graphs. We give combinatorial characterizations of
componentwise linearity of $I(D)$ if $V^+$ are sinks or $\vert V^+ \vert\leq
1$. Furthermore, if $G$ is chordal or bipartite or $V^+$ are sinks or $\vert
V^+ \vert\leq 1$, then we show the following equivalence for $I(D)$: $$
\text{Vertex splittable}\,\, \Longleftrightarrow\,\, \text{Linear quotient}\,\,
\Longleftrightarrow\,\, \text{Componentwise linear}.$$
