We calculate the production of a photon and two jets at forward rapidity in proton-nucleus collisions, within the hybrid dilute-dense framework in the Color Glass Condensate (CGC) formalism. After obtaining the cross section for both the quark-and gluon-initiated channels, we consider the correlation limit, in which the vector sum of the transverse momenta of the three outgoing particles is small with respect to the individual transverse momenta. In this limit, the cross section simplifies considerably and can be written in a factorized form, sensitive to various unpolarized and linearly-polarized transverse-momentum-dependent gluon distribution functions (gluon TMDs). Thus, we demonstrate for the first time that the emergence of a TMD factorization formula in the correlation limit, from CGC expressions, holds beyond the previously-considered simpler 2 → 2 processes.In the case of collinear factorization, there is a single hard scale, and the PDFs and FFs are only a function of this scale and of the longitudinal momentum fraction x of the parton. However, in less inclusive processes that depend on a second, smaller scale, these distributions need to be generalized to include the dependence on the transverse momentum of the parton, yielding the so-called transverse-momentum-dependent (TMD) PDFs and FFs. These TMD PDFs/FFs are of great interest since their measurement offers insight in the three-dimensional structure of the proton in terms of the QCD degrees of freedom. However, in contrast to their collinear counterparts, which are believed to be universal, they depend on the hard process under consideration and are therefore more complicated.The radiative corrections that are considered in collinear factorization involve logarithms ln Q 2 /µ 2 in the scale, and can be resummed with the help of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [1] evolution equations. In the TMD factorization framework [2], additional logarithms of the form ln Q 2 /k 2 t need to be resummed, where k t is the small transverse momentum scale in the problem.At very high energies, however, soft logarithms in the center-of-mass energy ln s are expected to be dominant, necessitating a resummation and subsequently an evolution equation in rapidity instead of the scale, provided by the (linear) Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [3]. BFKL involves a universal unintegrated gluon PDF, which just like the TMDs depends on transverse momentum. One of the most fascinating aspects of QCD at high energy or small x, is the phenomenon of saturation [4]: the damping of the steep rise of the gluon density predicted by BFKL, caused by gluon recombinations which become important due to the high density. The low-x evolution of the gluon density in the presence of these nonlinear interactions is described by an effective theory known as the Color Glass Condensate (CGC), which leads to the nonlinear BK-JIMWLK evolution equations [5][6][7][8].Within the CGC framework, the hybrid formalism [9] is used to study single inclusive particle produ...