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In this final part of a series of three papers, we will assemblesupersymmetric expressions for one-loop correlators in pure-spinor superspacethat are BRST invariant, local, and single valued. A key driving force in thisconstruction is the generalization of a so far unnoticed property at treelevel; the correlators have the symmetry structure akin to Lie polynomials.One-loop correlators up to seven points are presented in a variety ofrepresentations manifesting different subsets of their defining properties.These expressions are related via identities obeyed by the kinematicsuperfields and worldsheet functions spelled out in the first two parts of thisseries and reflecting a duality between the two kinds of ingredients.  Interestingly, the expression for the eight-point correlator following fromour method seems to capture correctly all the dependence on the worldsheetpunctures but leaves undetermined the coefficient of the holomorphic Eisensteinseries ${\rm G}_4$. By virtue of chiral splitting, closed-string correlatorsfollow from the double copy of the open-string results.

Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure