I'll show that space metric is a category.
Proof. Let $\mathfrak{G}$-structure and $\mathbb{X}=(X,d_{X})$ and $\mathbb{Y}=(Y,d_{Y})$ a space metrics, elements of $\mathfrak{G}$. let the continuous functions be the elements of Mor $Mor_{\mathfrak{G}}(\mathbb{X},\mathbb{Y})$.
I should show axiom of morphism:
Mor 1 Let $\mathbb{X}=(X,d_{X})$ , $\mathbb{Y}=(Y,d_{Y})$ and $\mathbb{Z}=(Z,d_{Z})$ spaces metric. if $f\in Mor_{\mathfrak{G}}(\mathbb{X},\mathbb{Y})$ and $g\in Mor_{\mathfrak{G}}(\mathbb{Y},\mathbb{Z})$. Suppose that $f:(X,d_{X}) \rightarrow (Y,d_{Y})$ is continuous at $a\in X$ and g is continuous at $f(a)$.
Let $\varepsilon>0 $, as $g$ is continuos at $f(a)$, there exist $\beta>0$ such that $d_{Z}(g(f(x)),g(f(a))<\varepsilon$ if $d_{Y}(f(x),f(a))<\beta$. Again $f$ is continuous at $a$, there exist $\delta>0$ such that $d_{Y}(f(x),f(a))<\beta$ if $d_{X}(x,a)<\delta$.
Hence $d_{Z}(g(f(x)),g(f(a))<\varepsilon$ if $d_{X}(x,a)<\delta$, which show that $gof$ is continuous. Hence $gof\in Mor_{\mathfrak{G}}(\mathbb{X},\mathbb{Z})$.
Mor 2.Let $f\in (X,d_{X})\rightarrow (X,d_{X})$- such that $f(x)=x$ for all $x\in X$ and I'll show that $f$ is continuous in $a\in X$.
Let $\varepsilon >0$, there exist $\delta=\varepsilon $, such that $d_{X}(x,a)<\delta$ implies $d_{X}(f(x),a)<\varepsilon$.
Hence $f\in Mor_{\mathfrak{G}}(\mathbb{X},\mathbb{X})$.
Hence $\mathfrak{G}$ is a category.
I am not sure if I am proving the necessary axioms, I have had doubt if it is time to prove the associative property, excuse my English.