uniform spaces if each uniform space $X_n$ is a subspace of the uniform space $X_{n+1}$, so the identity inclusion $X_n\to X_{n+1}$ is a uniform embedding.

The uniformity of the uniform direct limit ${\mathrm u\mbox{-}\kern-2pt\varinjlim}X_n$ of a tower $(X_n)_{n\in{\omega}}$ of uniform spaces was described in [@BaR] with help of uniform pseudometrics.

Let us recall that a pseudometric on a uniform space $Y$ is [*uniform*]{} if for every ${\varepsilon}>0$ the set $$\{d<{\varepsilon}\}:=\{(x,y)\in Y{\, ;\;}d(x,y)<{\varepsilon}\}$$ belongs to the uniformity ${\mathcal U}_Y$ of $Y$. By [@En 8.1.10], the uniformity ${\mathcal U}_Y$ of a uniform space $Y$ is generated by the family ${\mathcal{P\!M}}_Y$ of all uniform pseudometrics on $Y$ in the sense that the sets $\{d<1\}$, $d\in{\mathcal{P\!M}}_Y$, form a base of the uniformity ${\mathcal U}_Y$.

Let $(X_n)_{n\in{\omega}}$ be a tower of uniform spaces. A sequence of pseudometric $(d_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}$ is called [*monotone*]{} if $d_n\le d_{n+1}|X_n^2$ for every $n\in{\omega}$. Let $${\operatornamewithlimits{\bigwedge}}_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}=\{(d_n)_{n\in{\omega}}\in{\operatornamewithlimits{\textstyle{\prod}}}_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}{\, ;\;}(d_n)_{n\in{\omega}}\mbox{ is monotone}\}$$be the subspace of Cartesian product, consisting of monotone sequences of uniform pseudometrics on the uniform spaces $X_n$.

A family ${\mathcal A}\subset{\operatornamewithlimits{\bigwedge}}\limits_{n\in{\omega}}{\mathcal{P\!M}}_{X_n}$ is defined to be [*adequate*]{} if for each sequence of entourages $(U_n)_{n\in{\omega}}\in\prod_{n\in{\omega}}{\mathcal U}_{X_n}$ there is a monotone sequence of uniform pseudometrics $(d_n)_{n\in{\omega}}\in{\mathcal A}$ such that $\{d_n<1\}\subset U_n$ for all $n\in{\omega}$.

The following 