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We consider compression of higher-order solitons in asymmetric and symmetric dual-core fibers (couplers) by means of numerical simulations of the corresponding coupled nonlinear Schrödinger equations. We demonstrate that an asymmetric coupler with different dispersion coefficients in its two cores provides almost the same degree of compression for the soliton tunneling from the core with a larger dispersion into the one with a smaller dispersion as the single-core fiber with the same jump of the dispersion coefficients. The pedestal around the compressed soliton is smaller when using the coupler; however, the necessary compression length is larger for the coupler than for the single-core fiber. An advantage of using the coupler is that it allows one to avoid a junction between two pieces of the fiber with different dispersions, which is inevitable if one uses the single-core fiber. Next, we consider compression of a higher-order soliton in a symmetric coupler. We demonstrate that, using the so-called soliton effect, one can achieve a record compression ratio (approximately 20), which is larger than that for any other known soliton compressor, with a very small share of the total energy in the pedestal. The corresponding compression length is still larger than for the single-core fiber, but the compression quality is much better. Lastly, the symmetric coupler allows, in some cases, to split the incoming pulse into two nearly identical compressed pulses outcoming from the two cores.