Sensitivity analysis of coupled systems typically propagates derivatives of single-valued maps, whereas multiobjective optimization generally treats sensitivity at the system level rather than through interconnected subsystems. We develop a mathematical framework for compositional sensitivity analysis in multiobjective system design, building on monotone co-design. Subsystems are represented as design problems (DPs), interconnected through series, parallel, union, intersection, and feedback composition. We introduce tangential DPs (TDPs) to describe first-order variations around an operating point, show that they preserve the structure of DPs, and characterize their queries through graphical derivatives. To account for multiple internal realizations of a system-level operating point, tangential co-design problems compose families of local TDPs by taking a union over these realizations while retaining the original interconnection structure. We show that these compositions yield families of TDPs and establish conditions under which sensitivity analysis commutes with individual composition operations. Beyond infinitesimal sensitivity, we show that bounded tangential models of convex DPs provide tighter outer approximations of feasible finite changes, which can be propagated through co-design problems without union compositions. We conclude with several convex and nonconvex numerical examples that showcase the framework's accuracy and computational benefits.
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