Later, when we meet with professor Alex of the MI6...

Although you said that you were "acquainted" with the professor, why did you still need to introduce yourself?

Perhaps it was because you didn't want the professor to carelessly expose your true identity himself...

...that you had to illustrate a false identity on the spot?

Ah, that's right... asides from the MI5, the Home Office is also responsible for another organization.

That is, "The Police"

As fellow organizations belonging to the Home Office, the MI5 and the Police have a cooperative relationship...

It wouldn't be unusual if an MI5's investigator just happened to show up at a police's crime scene.