Categories of sets and functions can be considered as subcategories of **Set** but when considered as subcategories of the category **SubSet**, of pairs of sets with pairs $(X,S)$, $S\subseteq X$, as objects and functions $f:X\to Y$ with $f(S)\subseteq T$ as morphisms $(X,S)\to (Y,T)$, the morphisms of **SubSet** seems to reveal the form of the morphisms of the subcategories. Examples:

*Magmas*. $R\subseteq (X\times X)\times X$ (the composition). Morphisms are functions
$\alpha:(X\times X)\times X\longrightarrow(X^\prime\times X^\prime)\times X^\prime$ such that
$((x,y),z)\in R \Rightarrow \alpha((x,y),z)\in R^\prime$. Functions
$\alpha_1,\alpha_2,\alpha_3:X\longrightarrow X^\prime$ exists such that
$\alpha((x,y),z)=((\alpha_1(x),\alpha_2(y)),\alpha_3(z))$ and if $\alpha$ is such that $\alpha_1=\alpha_2=\alpha_3$, then $\alpha_1$ correspond to magma morphisms, $((x,y),z)\in R \Rightarrow ((\alpha_1(x), \alpha_1(y)),\alpha_1(z))\in R^\prime$. That is,
$xy=z\Rightarrow\alpha_1(x)\alpha_1(y)=\alpha_1(z)$.

*Topological spaces*. $\tau\subseteq \mathcal{P}(X)$. Morphisms are
functions $\mathcal{P}(X)\overset{\alpha}{\longrightarrow}\mathcal{P}(X^\prime)$ such that
$\mathcal{O}\in\tau \Rightarrow \alpha(\mathcal{O})\in \tau^\prime$. If there is a function
$f:X^\prime\longrightarrow X$ such that $\alpha = \mathcal{Q}(f)$, where $\mathcal{Q}$ is the contra-variant power set functor, this correspond to the condition
$\mathcal{O}\in\tau\implies f^{-1}(\mathcal{O})\in\tau^\prime$.

Hence, studying subcategories of **Set** yields that the morphisms are functions, while studying subcategories of **SubSet** also yields additional information of the morphisms. So my questions are:

Is this approach used in mathematics?

Has anyone published something about this?