$\{\alpha_i : A_i \to C_i \}$ family of maps $\implies \exists ! \alpha : \prod_i A_i \to \prod_i C_i$ such that $\pi_i^C \alpha \to \alpha_i \pi_i^A$

Question

Suppose that $\prod_i A_i, \prod_i C_i$ exist in a category, and that there is a family of maps $\{\alpha_i : A_i \to C_i\}$. There exists a unique $\alpha : \prod_i A_i \to \prod_i C_i$ such that $\pi_i^C \alpha = \alpha_i \pi_i^A \ \forall i$. Where $\pi_i$'s are the projections in the definition of product. What I've done is drawn a diagram of the definition of product:

_{(source: presheaf.com)}

But how do I come to the conclusion? I am not familiar with how to work with products. I want to be as non-concrete as possible in the proof.

Answer 1

You have morphisms $\alpha_j \circ \pi_j : \prod_i A_i \to C_j$. Apply the universal property of the product to get a morphism $\prod_i A_i \to \prod_i C_i$.