Which are the morphisms for sets with semimetrics? A semimetric is due to Wikipedia defined as a metric but without the triangle inequality. Intuitively it seems possible to define some kind of continuity for functions between such sets $f:A\to B$, a "small change" in input will result in only a "small change" of output:
$$\forall\epsilon>\!0\,\exists\delta>0: d(a,a')<\delta\implies d(f(a),f(a'))<\epsilon$$
but generally the triangle inequality is needed to show that the composition of two continuous functions is continuous.
So which are the morphisms in the category of sets with semimetrics?
 A: A modulus of continuity is a function $[0,\infty]\xrightarrow{\omega}[0,\infty]$ so that


*

*$r_1\leq r_2$ implies $\omega(r_1)\leq\omega(r_2)$.

*$\omega(0)=0$ and $\omega$ is continuous at $0$.


Fix an order-preserving function $[0,\infty]\times[0,\infty]\xrightarrow{\oplus}[0,\infty]$.
A $\oplus$-metric on a space $A$ is a function $A\times A\xrightarrow{d_A}[0,\infty]$ so that


*

*$d(a,a)=0$

*$d(a,c)\leq d(a,b)\oplus d(a,c)$


Notice that the case where $[0,\infty]\times[0,\infty]\xrightarrow{\oplus}[0,\infty]$ is identically $\infty$ is exactly the case of a "premetric space''.
If $A$ and $B$ are $\oplus$-metric spaces, then a function $A\xrightarrow{f}B$ has modulus of continuity $[0,\infty]\xrightarrow{\omega_a}[0,\infty]$ at $a\in A$ if


*

*$d_B(f(a),f(a')\leq\omega_a(d_A(a,a'))$


A morphism between such $\oplus$-metric spaces is then a function $A\xrightarrow{f}B$ equipped with a modulus of continuity $[0,\infty]\xrightarrow{\omega_{f,a}}[0,\infty]$ at every point $a\in A$. 
Note that with this definition


*

*uniformly continuous functions (which you defined) are morphisms that have the same modulus of continuity at every point.

*non-expansive maps (i.e. $d_B(f(a),f(a'))<d_A(a,a)$) are morphisms with modulus of continuity the identity function.

*continuous maps are morphisms that have forgotten what their moduli of continuity are.


As far as I know, category theorists have studied extensively the category of metric spaces and non-expansive maps between them because there $0$ is a two-sided identity for $\oplus=+$, and metric spaces end up being enriched categories with non-expansive functions corresponding to enriched functors.
