additive functor preserving biproducts preserves finite products

Question

Let $F:C \rightarrow D$ be a functor between two additive categories. The claim (p53, prop 40) is :

If $F$ preserves biproducts then $F$ preserves finite products.

For the proof the author wrote:

It suffices to prove that $F$ preserves zero object.

I do not get this argument. I thought there is nothing to prove and this follows by the characterization that for any product $(A\times B, \pi_1, \pi_2)$ we may extend this to a biproduct. $(A \times B, \pi_1, \pi_2, i_1, i_2)$.

So $F$ preserving biproduct implies preserving products (including initial and terminal objects)?

Answer 1

Preserving biproducts implies preserving products of pairs. An induction argument, shows that preserving terminal object and product of pairs implies preserving finite products.