Let $M=\{m_1,\dots\}$ be a set that contains several sets $m_i\in M$. I need to define a correspondence $\kappa$ that whenever inputed a set $m_i\in M$, returns me a subset of that particular set $m_i$. I do not know how to formalise this notion correctly.
My best attempt so far is this one: let $\kappa:M\rightrightarrows m_i$, where $m_i$ is the input of $\kappa$. This way of expressing such a simple idea seems unsatisfactory and imprecise: I need a better (more formal way) of specifying that "where $m_i$ is the input of $\kappa$". Could you please help me formalise it correctly?
I'm sorry for my rather unclear title, I hope somebody can understand my problem and provide some guidance.
You will not find any standard notation for this I believe. One thing you could do is define $\kappa$ as a function
$\kappa: M \rightarrow \Pi_i \mathcal{P}(m_i)$
where $\kappa(m_i) \in \emptyset \times ... \times \mathcal{P}(m_i) \times \emptyset \times ...$ and by $\mathcal{P}$ we mean the powerset. This way, such a function would uniquely to a set $m_i$ assign a subset of $m_i$ by checking its value of the $i$'th coordinate.