  We introduce the quiver of a bicomodule over a cosemisimple coalgebra.
Applying this to the coradical $C_0$ of an arbitrary coalgebra $C$, we give an
alternative definition of the Gabriel quiver of $C$, and then show that it
coincides with the known $\operatorname {Ext}$ quiver of $C$ and the link
quiver of $C$. The dual Gabriel theorem for a coalgebra with separable
coradical is obtained, which generalizes the corresponding result for a pointed
coalgebra. We also give a new description of $C_1$ of any coalgebra $C$, which
can be regarded as a generalization of the first part of the well-known
Taft-Wilson Theorem for pointed coalgebras. As applications, we give a
characterization of locally finite coalgebras via their Gabriel quivers, and a
property of the Gabriel quiver of a quasi-coFrobenius coalgebra.
