  The relationship between Jordan and Lie coalgebras is established. We prove
that from any Jordan coalgebra $\langle A, \Delta\rangle$, it is possible to
construct a Lie coalgebra $\langle L(A), \Delta_{L}\rangle$. Moreover, any dual
algebra of the coalgebra $\langle L(A), \Delta_{L}\rangle$ corresponds to a Lie
algebra that can be determined from the dual algebra for $\langle
A,\Delta\rangle$, following the Kantor--Koecher--Tits process. The structure of
subcoalgebras and coideals of the coalgebra $\langle L(A), \Delta_{L}\rangle$
is characterized.
