why coproduct of one and one is two and not three?

Question

Considering a category of finite sets, why is coproduct of 1 and 1, 2 and not 3? Specially in terms of universal property.

Answer 1

Why would it be $3$ ?

Let $i_1, i_2: \{0\}=1\to \{0,1\}=2$, $i_1(0) =0, i_2(0)=1$ Let $f,g: 1 \to X$ be two maps.

Let $[f,g] : 2\to X$ defined by $[f,g](0) = f(0), [f,g](1)= g(0)$.

Then clearly, $[f,g]\circ i_1 = f, [f,g]\circ i_2 = g$. Moreover, clearly, a map such that these two equations hold must be defined in the same way, so that $[f,g]$ is the unique map making the appropriate diagram commute.

Therefore $2$ is a coproduct of $1$ and $1$. But $3$ is not isomorphic to $2$ in $Set$, and therefore it cannot be another coproduct.