solution verification – Basic proof Group theory

Let F be the set of bijective functions 𝑓 ∶ 𝐴 → 𝐴. Is 𝐹 under the composition operations
◦ form a Group (structure similar to ℝ with +)?

Let $F=lbrace f:Ato A mid f, text{is a bijection}rbrace$ let us consider the operation $circ: Atimes A to A$ the usual composition of functions, we should prove that $(F,circ) $ is a group.

$i)$ There are the identity function $id:Ato A$ which is bijective such that
$idcirc f=f circ id=f$.

$ii)$ Notice that if $f,g,hin F$ since $f,g,h$ are bijective then
$(fcirc g)circ h= fcirc (g circ h)$ and as requierd the composition of bijective functions is also bijective.

$iii)$ For each function $fin F$ since $f$ is bijective there are $gin F$ such that $fcirc g=gcirc f= id_A$
Is it correct?