This paper explores the symmetry descent procedure in quantum field theories (QFTs), focusing on how the symmetry structure of a d-dimensional QFT can be encoded in a (d+1)-dimensional topological field theory (SymTFT). The SymTFT is constructed by considering the boundary conditions of a higher-dimensional topological field theory, which captures the local dynamics of the original QFT. The paper clarifies the subtle aspects of this construction, particularly in the context of supersymmetric QFTs arising from string theory or M-theory on conical singularities. The key idea is that the SymTFT can be obtained by reducing the topological sector of the higher-dimensional theory on the base of the cone, leading to a description of the original QFT's symmetries. The paper also discusses the BF sector of the SymTFT, which arises from integrating an auxiliary theory in one higher dimension. This BF sector is crucial for understanding the non-commutativity of certain symmetries in the QFT, such as 1-form and 2-form symmetries, which are related to fluxes in the higher-dimensional theory. The paper also addresses the challenge of constructing actions for self-dual fields, which is closely related to the problem of deriving the BF action. It uses a strategy initiated by Witten, where the partition function of a self-dual field is defined in terms of Chern-Simons theory on a manifold with boundary. This approach allows for the emergence of self-dual degrees of freedom on the boundary manifold. The paper further discusses the use of differential cohomology to describe the local dynamics of the QFT, including the integration of differential cochains and the treatment of gauge transformations. It provides examples of how these concepts apply to specific cases, such as U(1) gauge theories in two and even dimensions, and BF theories on manifolds with boundary. Overall, the paper provides a detailed analysis of the symmetry descent procedure, emphasizing the role of topological field theories in encoding the symmetry structure of QFTs and the challenges involved in deriving the BF sector and other topological aspects of the theory.This paper explores the symmetry descent procedure in quantum field theories (QFTs), focusing on how the symmetry structure of a d-dimensional QFT can be encoded in a (d+1)-dimensional topological field theory (SymTFT). The SymTFT is constructed by considering the boundary conditions of a higher-dimensional topological field theory, which captures the local dynamics of the original QFT. The paper clarifies the subtle aspects of this construction, particularly in the context of supersymmetric QFTs arising from string theory or M-theory on conical singularities. The key idea is that the SymTFT can be obtained by reducing the topological sector of the higher-dimensional theory on the base of the cone, leading to a description of the original QFT's symmetries. The paper also discusses the BF sector of the SymTFT, which arises from integrating an auxiliary theory in one higher dimension. This BF sector is crucial for understanding the non-commutativity of certain symmetries in the QFT, such as 1-form and 2-form symmetries, which are related to fluxes in the higher-dimensional theory. The paper also addresses the challenge of constructing actions for self-dual fields, which is closely related to the problem of deriving the BF action. It uses a strategy initiated by Witten, where the partition function of a self-dual field is defined in terms of Chern-Simons theory on a manifold with boundary. This approach allows for the emergence of self-dual degrees of freedom on the boundary manifold. The paper further discusses the use of differential cohomology to describe the local dynamics of the QFT, including the integration of differential cochains and the treatment of gauge transformations. It provides examples of how these concepts apply to specific cases, such as U(1) gauge theories in two and even dimensions, and BF theories on manifolds with boundary. Overall, the paper provides a detailed analysis of the symmetry descent procedure, emphasizing the role of topological field theories in encoding the symmetry structure of QFTs and the challenges involved in deriving the BF sector and other topological aspects of the theory.