This argument is almost there. The theorem you mentioned about the quotient requires that you are taking a topological quotient on the left of your diagram, whi

This argument is almost there. The theorem you mentioned about the quotient requires that you are taking a topological quotient on the left of your diagram, which prescribes a specific topology for the quotient space[1] (which you have not demonstrated is the topology you are using). Let Q1: X -> Y. This is the topology such that the open sets for Y are precisely those for which their preimage via Q1 is open in X. If you think this sounds a lot like continuity, you're right, but it adds one crucial detail: continuity only requires that whatever open sets we have for Y, their preimage is open (so for example you can make Y have the topology of a point, and Q1 would still be continuous). For quotient maps, we require that any set in Y with an open preimage be open, which fixes exactly one topology. This makes sense, because otherwise the quotient could pick from a laundry list of (non-isomorphic) topologies which keep the quotient map continuous, breaking the topological invariance of the quotient map. [1]: https://en.wikipedia.org/wiki/Quotient_space_(topology) https://en.wikipedia.org/wiki/Quotient_space_(topology)