The set theoretic Cartesian product creates an "output" set from a set of "input" sets, so that each member of the output set corresponds to the selection of one element each in every input set. There is also a standard Cartesian product of graphs and a standard Cartesian product of functions.
My question is, if I am learning a new subfield of mathematics and I see a "Cartesian product" analog, then what properties should I expect it to have? My naive intuition would be that it should change a list of objects that have sizes $n_1, n_2, \dots , n_k$ into a new object that has size $n_1 \cdot n_2 \cdots n_k$ and that to learn exactly the definition I would have to carefully read how it is defined. Is this intuition misleading or incomplete? Does the concept of "Cartesian product" have a technical meaning that is as wide in scope as it is used in various subfields, or would something like category theory be required for such a meaningful generalization?