Existence of a map via universal property?

Question

Let $X$ be a topological space and $A,B$ be subspaces of $X.$ Let $C \subseteq A \cap B.$ Let $f : A \longrightarrow B$ be a map. Then does it necessarily induce a map $\widetilde {f} : A/C \longrightarrow B/C\ $? If not, then what are the criterions the map $f$ should satisfy to induce the map $\widetilde f\ $?

EDIT $:$ I think if $f$ maps $C$ homeomorphically onto $C$ then we can have such maps by universal property of quotient topology. In fact then the map will be one-one and onto.

Answer 1

We have the canonical quotient maps $\pi_A: A\to A/C$ and $\pi_B :B\to B/C$. From $f$ we get a map $\pi_B\circ f : A\to B/C$. This induces a map $A/C\to B/C$ if, and only if, $\pi_B\circ f$ is constant on $C$. We see that this is the case if, and only if, one of the following holds :

1. $f|_C$ is constant ;

or

2. $f(C) \subset C$