Like many booksellers Lackington and co also published books, including Lackington’s own memoirs.
Lackington, Allen, & Co. began publishing in 1794, with the republication of Issac Watts’ The Improvement of the Mind and Arabian Nights (this last was published along with a number of other publishers. In the following years the company mainly published classics of English (like Samuel Johnson and Thomas Sheridan) or in translation (like Petrarch, Cicero, and Longinus).
The company of Lackington, Hughes, Harding, Mavor, & Jones was the publisher of Mary Shelley’s Frankenstein in 1818; they only printed a short run of 500 copies, which is tiny when you think about the number of volumes the Temple of the Muses stocked and sold. There’s a nice post on this at the University of Iowa Library site.