abstract algebra – The big picture regarding products and coproducts in categories

I am new to category theory. I have been reading about products and coproducts in Chapter 0 of Topology: A Categorical Approach by Bradley, Bryson and Terilla.

Is it fair to say that products and coproducts are really just methods for putting two (or more) objects from a category together to create a new object within the same category?

I am trying to grasp the essence of what’s going on before worrying about all the details (Cartesian product, direct product, direct sum, free product, disjoint union, etc.). I appreciate any help.