nLab
family of sets

A family of sets consists of an index set $I$ and, for each element $k$ of $I$, a set $S_k$.

Given $I$, a set $F$ and a function $p:F\to I$, we get a family of sets by defining $S_k$ to be the preimage $p^{*}(k)$.

Conversely, given a family of sets, let $F$ be the disjoint union

$$\underset{k}{⨄}{S}_k=\{(k,x)\mid k\in I,x\in S_k\}$$\biguplus_k S_k = \{ (k,x) | k \in I, x \in S_k \}

and let $f(k,x)$ be $k$.

(We should talk about ways to formalise this concept in various forms of set theory and when the latter construction above requires the axiom of collection.)