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If you already have postfix properly set up and running, edit
/etc/postfix/main.cf and append this:
smtp_sasl_auth_enable = yes relayhost = smtp.mailgun.org smtp_sasl_security_options = noanonymous smtp_sasl_password_maps = hash:/etc/postfix/sasl_passwd
Then add your smtp credential of mailgun to postfix:
echo 'smtp.mailgun.org email@example.com:password' > /etc/postfix/sasl_passwd
Protect the file and populate it:
chmod 600 /etc/postfix/sasl_passwd postmap /etc/postfix/sasl_passwd
And finally restart postfix:
service postfix restart
Now your server should send its mails via mailgun. Just give it a test shot!