Origin of 'The Lesson’ monogram
Apprentice Renoir, unobtrusively placed his initials in the hairline of the porcelain paintings of Empress Marie Louise. It amused him that M.Levy, owner of Levy – Freres at Rue du Temple Paris, didn’t see his 'initialling technique’ (1).

Renoir’s portrait appears to be an 'occasion’ for Jeanne.
Her hair has been prepared and her attire smartly presented.
The background indicates it was completed in the same room.
References:
(1) Initialling technique sourced in a publication on Impressionism.
Renoir My Father by Jean Renoir (Little, Brown & Company) cites anecdotal history of M.Levy’s handling of the apprentice Renoir from 1854 to 1858.
P60 – 61 Renoir’s profiles of Marie Antoinette were so successful, Mme.Levy had to persuade M.Levy to allow him to paint other historic portraits.
P68 – 69 Having studied the works of Watteau and Boucher at the Louvre Museum
’I longed to copy them on my porcelain…… I tried in vain to convince him’.
(2) If Renoir viewed The Footbridge, he will have seen the design of Sisley’s monogram.
See pages: The Footbridge Monogram, Origin of the Footbridge Monogram.