Proving a map in a commutative diagram is continuous.

A map defined on a quotient space $G{/}H$ is continuous iff its composition with the quotient map $p : G \to G{/}H$ is continuous, so $h$ is continuous iff $ h \circ p$ is, and $h \circ p = p \circ f$ by the diagram, and $p$ is continuous by definition, so $h$ is continuous when $f$ is (so we have a composition of continuous maps in $p \circ f$).