Quotient topology seems to satisfy the universal property for coproducts at first glance. However, at second glance they seem to fail to fit into that frame in general since not every map passes to the quotient. So, do I miss some argument or is it simply that the quotient topology belongs to another categorical notion?
To be more precise:
The quotient topology satisfies a universal property in the sense:
A surjective map from some initial topological space induces a unique topology on its codomain s.t. whenever theres a map from the quotient space to some arbitrary topological space then it is continuous iff its composition with the quotient map is continuous. ... But this is somehow stated not in the right direction as it is given for coproducts, i.e. given a map rather from the initial topological space there exists a unique continuous map from the quotient space. This fails to exist iff the map is not constant on fibres.