We put forward the hypothesis that the weak W boson be a compound of two 2-component Lorentz spinors. The resulting novel gammaWW vertex is no gauge field structure. Nevertheless, the Born amplitude of gammagamma --> W(L)W(L) respects partial-wave unitarity. As in the Yang-Mills case, the amplitude consists of a direct term, a crossed term, and a seagull term, and no unobserved particles are to be involved to get the ''good'' high-energy behavior. This is due to an imaginary pseudoscalar gammaWW interaction term. Significant differences between angular distributions and total cross sections of the non-Abelian case and the case of the composite bosons are displayed. The unitarity constraint applied to the reaction gammagamma --> W(T)W(T) leads to the prediction of the existence of a composite charged weak scalar PHI+/-. It constitutes the spin 0 state of the constituents forming W+/-. Furthermore, the existence of a second and heavy scalar-vector pair omega-X is predicted. These weak boson states are found to exclude the presence of a seagull graph. In the threshold region, the total cross section of gammagamma --> WW in the compositeness case is smaller than in the non-Abelian case. In a broad intermediate energy region it can be larger. Upper unitarity mass-bounds are estimated. They suggest m(PHI) almost-equal-to mw so that PHI+/- might be discovered by forthcoming experiments. The structure of the gammaPHIW, gammaXW and gammaomegaW transition vertices can be inferred without making recourse to unitarity. However, unitarity requires that the mass relation m(PHI)/m(W) = m(omega)/m(X) be valid.