  The integrated perturbation theory (iPT) is a set of methods in nonlinear
perturbation theory for the structure formation in the Universe. In Papers I
and II [arXiv:2210.10435, arXiv:2210.11085], the basic formalism and technical
methods of the iPT for cosmological tensor fields are developed, generalizing
the corresponding theory for scalar fields. In previous papers, methods to
predict statistical quantities, such as power spectra, correlation functions,
etc., of three-dimensional tensor fields are developed based on the iPT.
However, observations of tensors, such as angular momenta and shapes of
galaxies, etc., are only possible after the three-dimensional tensors are
projected onto the two-dimensional sky. In this paper, power spectra and
correlation functions of projected two-dimensional tensors are related to those
of original three-dimensional tensors, so that one can make predictions for the
observable statistics of projected tensor fields from the iPT. The relations
are consistently represented on the basis of irreducible decomposition of both
two- and three-dimensional tensors.
