Is this proof about composition functions correct?

Question

For this to work i did draw diagrams and i came up with example of functions.

Let F be the sets of all functions from the set of integers to set of integers. Prove or disapprove:

FALSE

$\forall f,g,h \in F$ if $ f\circ h = g \circ h$ then $f=g$

Now, i prove the negation is true. Let $f,g,h$ be the functions from set of integers to set of integers defined as

$f=\{(3,1),(4,2),(5,2)\}$

$g=\{(3,2),(4,2),(5,2)\}$

$h=\{(1,3),(2,5)\}$

$ f\circ h = \{(1,1),(2,2)\}$

$g \circ h = \{(1,1),(2,2)\}$

We see that $ f\circ h = g \circ h$ . Also, $f\not= g$ because $\{(3,1),(4,2),(5,2)\} \not= \{(3,2),(4,2),(5,1)\} $ .

In other words, for input in function $f$ there is an output that has to be equal for function $g$ if the input given is that of $f$.

Answer 1

$f=\{(0,7),(1,4)\}$

$g=\{(0,7),(1,5)\}$

$h \equiv 0$ disproves.