  We perform a thorough first-principles study on superconductivity in yttrium
carbide halide Y$_2$$X_2$C$_2$ ($X$=Cl, Br, I) whose maximum transition
temperature ($T_{\rm c}$) amounts to $\sim$10 K. A detailed analysis on the
optimized crystal structures reveals that the Y$_2$C$_2$ blocks are compressed
uniaxially upon the halogen substitution from Cl, Br to I, contrary to the
monotonic expansion of the lattice vectors. With a nonempirical method based on
the density functional theory for superconductors within the conventional
phonon mechanism, we successfully reproduce the halogen dependence of $T_{\rm
c}$. Anomalously enhanced coupling of one C$_2$ libration mode is observed in
Y$_2$I$_2$C$_2$, which imply possible departure from the conventional pairing
picture. Utilizing the Wannier representation of the electron-phonon coupling,
we show that the halogen electronic orbitals and ionic vibrations scarcely
contribute to the superconducting pairing. The halogen dependence of this
system is hence an indirect effect of the halogen ions through the uniaxial
compressive force on the superconducting Y$_2$C$_2$ blocks. We thus establish a
quantitatively reliable picture of the superconducting physics of this system,
extracting a unique effect of the atomic substitution which is potentially
applicable to other superconductors.
