In Topics in Geometric Group Theory, section 3.C lemma 42,
For a finitely-generated group T, the following two properties are equivalent:
(i) T has uncountably many normal subgroups,
(ii) T has uncountably many non-isomorphic quotients.
In proof, it assumes that the cardinality of the set of normal subgroup of T is uncountable and the cardinality of the set of isomorphism classes of quotient groups of T is countable to obtain a contradiction. it shows that there exist an uncountably many homomorphisms from T onto quotient group Q. but the cardinality of the set of Q is countable as above assumption, it is contradiction.
Here is my question. In the last part of the proof, it has to be proven first that the group homomorphisms are injective, or If two quotients groups of finitely generated group are isomorphic, then corresponding normal subgroups are isomorphic. is it correct? if not, tell me why it is contradiction specifically please.