coproduct
The coproduct is the mirror image of the product, obtained by reversing every arrow. Where a product is the universal object you can map into, a coproduct is the universal object you can map out of — it bundles two objects together so that to define a map from the bundle you only need to define a map from each piece separately. Intuitively it is the most economical way of “putting A and B side by side” with no interaction forced between them.
Formally, a coproduct of A and B is an object A ⊔ B with two inclusion morphisms i : A -> A ⊔ B and j : B -> A ⊔ B, universal in the dual sense: for every object X with morphisms f : A -> X and g : B -> X, there is a unique morphism [f, g] : A ⊔ B -> X with [f, g] ∘ i = f and [f, g] ∘ j = g. This is exactly the product taken in the opposite category, which is why the two notions are perfect mirror images.
Despite the symmetric definition, coproducts look very different across categories. In Set the coproduct is the disjoint union; in Vect and in abelian groups it coincides with the product (the direct sum) for finitely many factors; in Grp it is the free product, a genuinely large and noncommutative object; in commutative rings it is the tensor product over Z. The lesson is that duality is formal but the concrete answers are not.
The coproduct of two copies of Z in Grp is the free group on two generators, F_2 = Z * Z, which is infinite and nonabelian. In abelian groups, by contrast, the coproduct of Z and Z is the direct sum Z ⊕ Z, abelian and equal to the product.
Reversing arrows flips product to coproduct, but answers diverge wildly.