  For a Tychonoff space $X$, the lattice $\mathscr U_X$ was introduced in L.A.
P\'erez-Morales, G. Delgadillo-Pi\~n\'on, 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 $\mathscr U_X$. To
be specific, the present paper deals, in its first half, with structural and
categorical properties of $\mathscr U_X$, while in its second part focuses on
cardinal characteristics of the lattice and how these relate to some cardinal
functions of the space $X$.
