Does THE library refer to the personal library of Colonel Anderson?

Does THE library refer to the personal library of Colonel Anderson?

The passage below is from The Library: A Fragile History by Andrew Pettegree.

For the fifteen years after the opening of the Dunfermline library, Carnegie’s involvement was largely limited to other places with which he had a close personal connection, including a workers’ library for one of his steel mills. When the city fathers of Pittsburgh rebuffed his first offer, unwilling to commit the matching funds on which Carnegie insisted, he turned instead to Allegheny City, Pennsylvania, where he had famously devoured books as a teenager in the personal library of Colonel Anderson. This cathedral of books, in a strange mock-Romanesque style, reflected the embryonic state of American library design. When in 1889 the chastened Pittsburgh council invited Carnegie to renew his offer, he bore no grudges. The library was incorporated into the Carnegie Institute on which he lavished $5 million.

Does THE library refer to the personal library of Colonel Anderson?

Thanks in advance.