On the Fundamental Groups of Products of Topological Spaces

The following proposition is true (1) Let G, H be non empty groupoids and x be an element of ∏ 〈G,H〉. Then there exists an element g of G and there exists an element h of H such that x = 〈g, h〉. Let G1, G2, H1, H2 be non empty groupoids, let f be a map from G1 into H1, and let g be a map from G2 into H2. The functor Gr2Iso(f, g) yields a map from ∏ 〈G1, G2〉 into ∏ 〈H1,H2〉 and is defined by the condition (Def. 1). (Def. 1) Let x be an element of ∏ 〈G1, G2〉. Then there exists an element x1 of G1 and there exists an element x2 of G2 such that x = 〈x1, x2〉 and (Gr2Iso(f, g))(x) = 〈f(x1), g(x2)〉. The following proposition is true