  We study the dinucleon (dineutron and diproton) correlation of the ground
states of $^9$Li, $^{10}$Be, and $^{9,10}$C. We assume an $\alpha+t$ core for
$^9$Li, an $\alpha+\alpha$ core for $^{10}$Be and $^{10}$C, and an
$\alpha+^3$He core for $^9$C, and investigate the effect of core structure
changes on the degree of dineutron formation and spatial expansion from the
core. For $^9$Li, $t$ cluster breaking in the core significantly enhances the
dineutron component inside the nuclei. Moreover, its component markedly depends
on the strength of the spin-orbit interaction since a dineutron is fragile and
dissociates readily due to the spin-orbit interaction. Compared with $^9$Li,
the dineutron of $^{10}$Be dissociates largely due to the stronger spin-orbit
attraction from the $\alpha+\alpha$ core than the $\alpha+t$ core. We also
investigate diproton features in $^{9,10}$C, the mirror nuclei of $^9$Li and
$^{10}$Be, respectively, and compare them with the dineutron features of $^9$Li
and $^{10}$Be. No qualitative difference is observed between diprotons and
dineutrons in the degree of formation at the surface, but a quantitative
difference may exist in the degree of dinucleon-size change far from the core.
