A terminal object is not the product of two other objects

Question

Consider a category $C$ with a terminal object $T$ why is it true that $T$ cannot be the product of two other objects $A$ and $B$. I see that we have a unique arrow $s_A:A \to T$ and a unique arrow $s_B:B\to T$ and a unique identity arrow $id_T:T\to T$ so if $T$ were the product of $A$ and $B$ we would have two projections $p_A:T\to A$ and $p_B:T\to B$, hence the unicity of the arrow from $T$ to $T$ would imply that $id_T=s_A\circ p_A =s_B\circ p_B$ but then i don't see how to get a contradiction!

Answer 1

It depends how you read: "other". But this is false: consider in the category of sets that the product of any two singleton sets (terminal objects) is again a singleton set (a terminal object).

If $T\cong A\times B$ then instead try to show that both $A$ and $B$ are also terminal. Conversely if $A,B$ are terminal then try to show that their product is also terminal.

So if "other objects" excludes other terminal objects, then this statement is correct. More clearly:

A terminal object cannot be the product of a non-terminal object with another object