[Math] Prove that if $f \circ g = f\circ h$, then $g = h$

Question

Given three functions,

$$f\colon A\to B, g\colon C\to A,h\colon C\to A$$

Prove or disprove that if $f \circ g = f \circ h$, then $g = h$.

Answer 1

Here is a counterexample: \begin{align} g(1) & = 1 \\ g(2) & = 2 \\ g(3) & = 3 \\ \\ h(1) & = 1 \\ h(2) & = 3 \\ h(3) & = 2 \\ \\ f(1) & = 1 \\ f(2) & = 2 \\ f(3) & = 2 \end{align}