The mechanical properties of cells and the extracellular environment they reside in are governed by a complex interplay of biopolymers. These biopolymers, which possess a wide range of stiffnesses, self-assemble into fibrous composite networks such as the cytoskeleton and extracellular matrix. They interact with each other both physically and chemically to create a highly responsive and adaptive mechanical environment that stiffens when stressed or strained. Here we show that hybrid networks of a synthetic mimic of biological networks and either stiff, flexible and semi-flexible components, even very low concentrations of these added components, strongly affect the network stiffness and/or its strain-responsive character. The stiffness (persistence length) of the second network, its concentration and the interaction between the components are all parameters that can be used to tune the mechanics of the hybrids. The equivalence of these hybrids with biological composites is striking.