The answer is cortex. Milady states directly that the cortex gives hair its strength, flexibility, elasticity, and shape — the middle of the hair's three structural layers.
The cuticle is the outermost, protective layer (not the source of strength or color), and the medulla is the innermost layer, sometimes absent entirely in fine hair. Natural hair color specifically comes from melanin pigment granules held within the cortex, reinforcing why this layer, not the others, is the answer.
Milady Standard Barbering, 6th ed., Ch. 17 (Chemical Texture Services), p. 585: "Cortex: The cortex gives the hair its strength, flexibility, elasticity, and shape."
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