Compressors, Gas Turbines and Jet Engines
An ideal air compressor cycle with clearance on P-V diagram can be represented by following processes

Two adiabatic, one isobaric and one constant volume
One adiabatic, one isobaric and two constant volumes
One adiabatic, two isobaric, and one constant volume
Two adiabatic and two isobaric

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

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