Scott - I believe your conjecture!

I think when you're saying 'has all coproducts' you mean what most category theorists when they say 'has binary coproducts': each pair of objects \\(x\\) and \\(y\\) has a coproduct, written \\(x + y\\). We can try to take the coproduct of any number of objects, even an infinite number, but binary coproducts are probably the most important case.

People often study categories with binary coproducts and an **initial object**, often written \\( 0\\): that's an object that has only one morphism to _any_ other object in the category.

An initial object can be seen as a 'nullary coproduct', and we can build up ternary, etc. coproducts from binary ones, so a **category with finite coproducts** is the same as a category with binary coproducts and an initial object.

Categories with finite coproducts are very nice: perhaps the most important is the category \\(\textbf{FinSet}\\) with finite sets as objects, and functions as morphisms.

I think when you're saying 'has all coproducts' you mean what most category theorists when they say 'has binary coproducts': each pair of objects \\(x\\) and \\(y\\) has a coproduct, written \\(x + y\\). We can try to take the coproduct of any number of objects, even an infinite number, but binary coproducts are probably the most important case.

People often study categories with binary coproducts and an **initial object**, often written \\( 0\\): that's an object that has only one morphism to _any_ other object in the category.

An initial object can be seen as a 'nullary coproduct', and we can build up ternary, etc. coproducts from binary ones, so a **category with finite coproducts** is the same as a category with binary coproducts and an initial object.

Categories with finite coproducts are very nice: perhaps the most important is the category \\(\textbf{FinSet}\\) with finite sets as objects, and functions as morphisms.