If a category has binary products does that implies that it has all finite products except for possibly the empty product?

Question

If a category has binary products does that implies that it has all finite products except for possibly the empty product?

My intuition says yes, because I would think that for $n \ge 2$ we can succesively take binary products.

Answer 1

That is correct.

If you want to be precise about it, then you can prove by induction on $n \ge 1$ that an the $n$-fold iterated binary product $(\cdots ((A_1 \times A_2) \times A_3) \times \cdots ) \times A_n$ of objects $A_1,A_2,\dots,A_n$, together with the 'obvious' projection maps, is an $n$-fold product in the usual sense.

...and then the category (with binary products) has all finite products if and only if it has a terminal object.