I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in which arrows are in opposite direction (as compared from one to other).
However, I didn't get any idea how to arrive at the universal property of direct sum and product. I mean, given the two definitions of direct sum/product - set theoretic description and universal property - I understood their equivalence. But, what is intuition for obtaining universal property from set theoretic description/definition.
Most of the books put these two definitions, show their equivalence, but no one give intuitive idea for obtaining universal property.
Edit [Added OP's comment]: To be more precise, suppose on one page, I wrote definitions of the direct sum and product in terms of set theory, and on another page, I wrote definitions in terms of universal property, then how can I solve the problem: which definition on page 1 is equivalent to exactly one definition on page 2?