When is a space homeomorphic to a quotient space?

Is the following theorem true? It seems straightforward but I haven't seen it published anywhere, not even as a corollary, so I'm concerned I've missed something. Discussions that introduce quotient spaces all seem to dance around this very simple and useful fact. Why don't they just come right out and say it?

Let $X$ and $Y$ be a topological spaces. Let $\sim$ be an equivalence relation on $X$. Then $Y$ is homeomorphic to the quotient space $X/{\sim}$ iff there exists a quotient map $f:X \to Y$ that induces the same partition as $\sim$.

It is well known (and stated in most textbooks) that a continuous surjection $f : X \to Y$ is a quotient map if and only if the following is satisfied for all functions $g : Y \to Z$:

$g$ is continuous if and only if $g \circ f : X \to Z$ is continuous.

An obvious corollary is this.

Given two quotient maps $f : X \to Y, f' : X \to Y'$ and a bijection $\phi : Y \to Y'$ such that $\phi \circ f = f'$. Then $\phi$ is a homeomorphism.

Each quotient map $f : X \to Y$ induces an equivalence relation on $X$ by defining $x \sim x'$ iff $f(x) = f(x')$. Now apply the corollary to $f$ and the quotient map $p : X \to X/\sim$.

In fact, this does not frequently occur as an explicit statement in the literature.

I just learned that it occurs in Munkres. See Xiang Yu's answer to How does the quotient $\mathbb{R}/\mathbb{Z}$ become the circle $S^1$?.