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

The compression in both the cylinders is polytropic
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

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

Isothermal h.p. to the BHP of motor
Work to compress air isothermally to work for actual compression
Isothermal h.p. to adiabatic h.p.
Power to drive compressor to isothermal h.p.

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