Pressure Change in Adiabatic v. Isothermal Processes

[justadream]

Why does pressure change more quickly with volume in adiabatic processes than isothermal processes?

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[The Brown Knight]

My understanding is that in an adiabatic expansion of a gas, the gas cools (since internal energy decreases since the gas exerts work on the surroundings but no heat is added) whereas in an isothermal process the gas stays the same temperature (internal energy doesn't change since heat is added into the system from the surroundings as the gas is expanding).
Because the gas is simultaneously increasing in volume and decreasing in temperature, adiabatic processes have more rapid changes in pressure with respect to volume.