  The significance of $\Lambda\Lambda$-$\Xi$N coupling in double-$\Lambda$
hypernuclei has been studied. The Pauli suppression effect due to this coupling
in $^6_{\Lambda\Lambda}$He has been found to be 0.43 MeV for the coupling
strength of the NSC97e potential. This indicates that the free-space
$\Lambda\Lambda$ interaction is stronger by about $5^{\circ}$ phase shift than
that deduced from the empirical data of $^6_{\Lambda\Lambda}$He without
including the Pauli suppression effect. In $^5_{\Lambda\Lambda}$He and
$^5_{\Lambda\Lambda}$H, an attractive term arising from $\Lambda\Lambda$-$\Xi$N
conversion is enhanced by the formation of an alpha particle in intermediate
$\Xi$ states. According to this enhancement, we have found that the
$\Lambda\Lambda$ binding energy ($\Delta B_{\Lambda\Lambda}$) of
$^5_{\Lambda\Lambda}$He is about 0.27 MeV larger than that of
$^6_{\Lambda\Lambda}$He for the NSC97e coupling strength. This finding deviates
from a general picture that the heavier is the core nucleus, the larger is
$\Delta B_{\Lambda\Lambda}$.
