Multi-spool compression systems are characterized by two or more compressor stages running at different rotational speeds. The response of an individual component can be different from an integrated system. Limiting operating conditions such as choke and stall points could have substantially different effects. The present paper explores the interactions and coupling significance between different stages of a multi-spool compression system. Further, an attempt is made by modifying the shape of the inter-compressor duct (ICD) to improve the system performance. The multi-spool system in this study comprises of the NASA stage 67 as the fan followed by in-house core and bypass ducts and a single stage booster. It is observed that the flow pattern in an ICD is entirely different in stand-alone modeling than in the integrated system modeling, owing to fan wakes and booster upstream influences. The booster performance is dependent on the duct exit flow pattern. The shape of the baseline ICD is tailored to reduce extra losses which is generated due to reduction in the length of the ICD and hence making the system more compact. It is shown that the shape tailoring optimization of ICD done independently result in a significant improvement in the duct exit flow pattern and hence an improvement in the booster performance. However, this gain in the performance is reduced to marginal values for an integrated system. This happens due to a strong coupling of the ICD flow pattern with the fan wakes and highly three dimensional nature of the ICD flow pattern. Therefore, it is found that component level optimization may not give rise to an equivalent system-level improvement.