From https://www.mimuw.edu.pl/~tarlecki/teaching/ct/papers/chap1.pdf , an excerpt from an educational book about universal algebra, category theory etc, from the section that talks about many-sorted sets:
Exercise 1.1.3. Extend the above definitions of union, intersection, product and disjoint union to operations on $I$-indexed families of $S$-sorted sets, for an arbitrary index set $I$. For example, the definition for product is $\left(\prod{\left\langle X_i\right\rangle_{i\in I}}\right)_s=\left\lbrace {f:I\rightarrow\bigcup_{i\in I}{\left(X_i\right)_s}|{f\left(i\right)\in\left(X_i\right)_s}}\text{ for all }i\in I\right\rbrace$ for each $s\in S$.
Weird. If I understand correctly, a cartesian product is defined here as a set of tuples (as expected), BUT a tuple is defined as a function that assigns a value to the index.
Problem is that, if I remember correctly, I've always been told something different: A function is defined as a set of ordered pairs (argument, value) such that no two pairs assign two different values two any given argument.
Since an ordered pair is a special case of a tuple... Do we have here a situation when an ordered pair is defined as a function while a function is defined as an ordered pair?
I suppose there may not be cyclic definitions in mathematics? Thus how are these two beasts (functions, tuples) typically defined?
It seems here that ordered pairs are already defined in this situation, as are functions, which are collections of ordered pairs. The definition is leveraging this to define the product over an arbitrary index set.
Ordered pairs are defined first, then functions. But in this case all that is behind us and we are defining something new. $I$ could have exactly two elements and hence the definition yields something resembling ordered pairs, though not identical. The new information is the specific index set.