empty object in category

I read Maclane's book recently.

• the product of empty product is terminal object.this make sense if use the universal definition.

in the category of finite products(this is equivalent to any two objects of the category has product).the book has said it had terminal object. Just take the product of two empty objects.

I am confused about the following:

-take two empty object,this means in category $C$of finite products,we can consider empty object as an object of $C$. this make sense because empty object do not destroy the structure of any category.

• but if we consider the category of preorder $Z$($i\le j$ iff there is only one arrow from $i$ to $j$),then it has terminal object $+\infty$ and initial object $-\infty$.is this true?
• I think you use the word "of" too often here. Are you sure you mean the category of finite products, rather than a category which has finite products? – Tobias Kildetoft Jul 10 '17 at 15:01
• Since $+\infty$ is not an object of that category, it simply cannot be the terminal object. – Mariano Suárez-Álvarez Jul 10 '17 at 15:09
• @TobiasKildetoft yeah ,my meaning is this. – Sky Jul 10 '17 at 15:10
• A terminal object of a catgory is an object of the category. Since $+\infty$ is not an object of $Z$, then it can be the terminal object of $Z$ just as much as my chair. – Mariano Suárez-Álvarez Jul 10 '17 at 15:15
• @Sky - As I answer below, $\mathbb{Z}$ has binary products (resp. coproducts), not all finite products (resp. coproducts). The nonexistence of a greatest (resp. least) element tells you exactly this. – Malice Vidrine Jul 10 '17 at 15:23

As Mariano states in the comments, $\mathbb{Z}$, as a preorder, has no initial or terminal object. $\mathbb{Z}$ does have binary products, and therefore products of arity $\geq 1$, but that's no guarantee that you get products of zero arity.