Questions about a topological category Given topological spaces $(X_i,\tau_i)$ with sets 
$\mathbf S_i=\{\mathcal A\in \tau_i^2|\mathcal A\supseteq\Delta X_i^2\}$, where $\tau^2$ is the product topology and $\Delta$ is the diagonal. The sets S make it possible to judge how close to points $x,y\in X$ are to each other: the smaller open sets $\mathcal A\in \mathbf S$ such that $(x,y)\in\mathcal A$, the nearer $(x,y)$ is to the diagonal and the nearer $x$ and $y$ are.
Define function $f:X_1\to X_2$ to be uniform iff:
$\;\forall x,y\in \!X_1\, \forall \mathcal E_2\!\in \mathbf S_2\, \exists\mathcal D_1\in\mathbf
S_1: ((x,y)\in\mathcal D_1\implies(f(x), f(y))\in\mathcal E_2)$
$\qquad$ or equivalent
$\;\mathcal E_2\!\in \mathbf S_2\implies((f\times f)^{-1}(\mathcal E_2)\subseteq \mathbf S_1 $.
Apparently, topological spaces $X$ with "uniform" functions form a category since the identity and the composition of "uniform" functions are "uniform" functions, and it would be interesting to know things about that category, for example if the inverse of a bijective morphism is a morphism.

It should be pointed out perhaps, that this category is not the
  category of uniform spaces and as far as I know the morphisms in
  general don't have to be continuous.

Further I wonder if "uniformity" is equivalent with
$\mathcal B\subseteq X_2\times X_2\implies((x,y)\in\overline{(f\times f)^{-1}(\mathcal B)}\implies (f(x),f(y))\in\overline{\mathcal B})$?
See also: Is this a criterion for continuity? about a similar but not equal condition on a function.
 A: The fonction $x \mapsto 1/x$ defined on $[1,+\infty[$ is uniformly continuous. (Exercise !) The function $x\mapsto 1/x$ defined on $]0,1]$ is continuous but not uniformly continuous (exercise again !) and these two functions are inverse one to each other.
As on a metric space, with the uniform structure given by the metric, being uniform amouts to be uniformly continuous, the previous toy example shows that the inverse of a bijective morphism is not necessarily a morphism in the category of uniform spaces with uniform maps as morphisms.
For you other question, the answer is yes, and I also think that it is a proposition in Bourbaki, Topologie générale, Chapitre II, or maybe an exercise in the exercises for this chapter, but I don't have it next to me, so that I cannot confirm.
EDIT : after the edit of the question, what is meant is not leading to a uniform structure indeed, and is definitely not a uniform structure as well.
The also from your see also is misplaced, as the definitions of uniformity are not the same here and there. This question has horribly bad natations (i,j etc) and is too much unprecise and changing.
