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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.

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    $\begingroup$ Welcome to MSE. What's the group operation on $X$? $\endgroup$
    – José Carlos Santos
    Nov 19, 2020 at 9:28
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    $\begingroup$ Surely the bijections of a set of three elements consists of the permutations of these three elements? [Many define a "permutation" in exactly this way.] $\endgroup$
    – ancient mathematician
    Nov 19, 2020 at 9:32
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    $\begingroup$ @JoséCarlosSantos $X$ is not a group, but $Bij(X)$ is a group for the composition law. $\endgroup$
    – TheSilverDoe
    Nov 19, 2020 at 10:08
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    $\begingroup$ @Laury Yes, you got it ! $\endgroup$
    – TheSilverDoe
    Nov 19, 2020 at 10:21
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    $\begingroup$ @TheSilverDoe Thank you very much ! $\endgroup$
    – Laury
    Nov 19, 2020 at 10:25

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