Isomorphic in Product Category

Question

For any family of two or more object $ A_{1}, A_{2},..., A_{n}$ in Category C prove that $ \prod_{i=1}^{n}A_{i} $ is isomorphic to $ \prod_{i=1}^{n-1}A_{i} \times A_{n} $?

Answer 1

Any cone with summit $Z$ over the discrete diagram $A_1, \cdots, A_n$ with legs $\lambda_1, \cdots, \lambda_n$ forms a cone over the discrete diagram $A_1, \cdots, A_{n-1}$ with legs $\lambda_1, \cdots, \lambda_{n-1}$, thus by the universal property of $\prod_{i = 1}^{n-1}A_i$, there exists a unique map $Z \to \prod_{i = 1}^{n-1}A_i$, this gives us a cone over the discrete diagram $\prod_{i = 1}^{n-1}A_i, A_n$. By the universal property of $\prod_{i = 1}^{n-1}A_i \times A_n$, there exists a unique map $Z \to \prod_{i = 1}^{n-1}A_i \times A_n$.

Therefore we have just shown that any cone over the discrete diagram $A_1, \cdots, A_n$ factors through $\prod_{i = 1}^{n-1}A_i \times A_n$. By the universal property of $\prod_{i = 1}^{n}A_i$, we have $\prod_{i = 1}^{n}A_i \cong \prod_{i = 1}^{n-1}A_i \times A_n$ as desired.