Compressors, Gas Turbines and Jet Engines
The overall isothermal efficiency of the compressor is defined as the ratio of

Isothermal power to the shaft power or B.P. of the motor or engine required to drive the compressor
Volume of free air delivery per stroke to the swept volume of the piston
Work required to compress the air isothermally to the actual work required to compress the air for the same pressure ratio
Isentropic power to the power required to drive the compressor

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Compressors, Gas Turbines and Jet Engines
The assumption made in two stage compression with inter-cooler is that

The suction and delivery of air takes place at constant pressure
All of the listed here
There is no pressure drop in the inter-cooler
The compression in both the cylinders is polytropic

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