In Mac Flecknoe by John Dryden, was Thomas Shadwell the same man as Mac Flecknoe?

The complete title of John Dryden' verse satire is "Mac Flecknoe; or, A satyr upon the True-Blew-Protestant Poet, T.S."

'T.S.' stands for Thomas Shadwell. So most obviously Mac Flecknoe stands for none other than Thomas Shadwell.

John Dryden satirizes Thomas Shadwell a famous playwright but a mediocre poet because of their serious disagreements over their critical opinions regarding contemporary drama and the genius and worth of Ben Jonson. Moreover, political differences also separated them: Shadwell was a Whig while Dryden was a supporter of the Stuart monarchy. To cap it all, Dryden was a Roman Catholic while Shadwell as the sub title of the poem indicates was a protestant.

To quote the e notes editor,

"The poem has been commonly adjudged the best short satiric poem in the English language."