morphism to product and commuting for each component

1. Proposition

Let 20231018-morphism_to_product_and_each_component_760824120891fc757bc0445145b027d09b237baa.svg be a category, a categorical product and morphisms

for each , we conclude that

2. Proof

2.1. 1) 2)

By assumption and universal property, there exists a unique morphism

20231018-morphism_to_product_and_each_component_2158d3dc77077c0a732e4202920b7cd8a2de08f5.svg

Thus 20231018-morphism_to_product_and_each_component_ff30c2997263865ed9ef266fbb601efe3848c80e.svg

2.2. 2) 1)

basic

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 18:43