New results regarding the lattice of uniform topologies on $C(X)$

For a Tychonoff space $X$, the lattice ${\mathcal{U}}_X$ was introduced in L.A. Pérez-Morales, G. Delgadillo-Piñón, and R. Pichardo-Mendoza, “The lattice of uniform topologies on $C(X)$”, Questions and Answers in General Topology, 39 (2021), 65-71. In the present paper we continue the study of ${\mathcal{U}}_X$. To be specific, the present paper deals, in its first half, with structural and categorical properties of ${\mathcal{U}}_X$, while in its second part focuses on cardinal characteristics of the lattice and how these relate to some cardinal functions of the space $X$.