On a new reformulation of Hadwiger's conjecture

Assuming that every proper minor closed class of graphs contains a maximum with respect to the homomorphism order, we prove that such a maximum must be homomorphically equivalent to a complete graph. This proves that Hadwiger's conjecture is equivalent to saying that every minor closed class of graphs contains a maximum with respect to homomorphism order. Let FF be a finite set of 2-connected graphs, and let CC be the class of graphs with no minor from FF. We prove that if CC has a maximum, then any maximum of CC must be homomorphically equivalent to a complete graph. This is a special case of a conjecture of Nešetřil and Ossona de Mendez.