Show that there is an injective homomorphism between $Aut(S_3)$ and $Bij(\{(12), (1 3), (2 3)\}$

Question

Let $X$ be equal to $\{(1 2), (1 3), (2 3)\}$. How can I prove the existence of an injective homomorphism between $Aut(S_3)$ and $Bij(X)$ ?

I was thinking about this :

Let $\varphi$ be a map between $Aut(S_3)$ and $Bij(X)$ defined as following :

For $\psi$ in $Aut(S_3)$, $\varphi(\psi) = f$ with $f : X \longrightarrow X$ such as $\forall x \in X$, $f(x) = \psi(x)$

Then I think I can show that $\varphi$ is an injective homomorphism.

Do you think it's correct ? Thank you for your help.