  We derive the scattering length of composite bosons (cobosons) within the
framework of the composite boson many-body formalism that uses correlated-pair
states as a basis, instead of free fermion states. The integral equation
constructed from this physically relevant basis makes transparent the role of
fermion exchange in the coboson-coboson effective scattering. Three potentials
used for Cooper pairs, fermionic-atom dimers, and semiconductor excitons are
considered. While the s-wave scattering length for the BCS-like potential is
just equal to its Born value, the other two are substantially smaller. For
fermionic-atom dimers and semiconductor excitons, our results, calculated
within a restricted correlated-pair basis, are in good agreement with those
obtained from procedures numerically more demanding. We also propose model
coboson-coboson scatterings that are separable and thus easily workable, and
that produce scattering lengths which match quantitatively well with the
numerically-obtained values for all fermion mass ratios. These separable model
scatterings can facilitate future works on many-body effects in coboson gases.
