The analysis of a text #2

The narrator recalls his childhood.
Mr. Wickfield is the master of his boarding house, one of his grand aunt's friends, and a lawyer.
Doctor Strong is the head master of the new school he goes to.
Doctor Strong had asked Mr. Wickiled to get a job for his young wife's cousin Mr. Maldon.
Today, Mr. Maldon has to get aboard a ship for India and he has gone after a farewell reception.

We walked very slowly home, Mr. Wickfield, Agnes and I, Agnes and I admiring the moonlight, and Mr. Wickfield scarcely raising his eyes from from the ground. When we, at last, reached our own door, Agnes discovered that she had left her little reticule behind. Delighted to be of any service to her, I ran back to fetch it.
I went into the supper-room where it had been left, which was deserted and dark. But a door of communication between that and the Doctor's study, where there was alight, being open, I passed on there, to say what I wanted, and to get a candle.
[David Copperfield by Charles Dickens]
I'd like to know if the underlined participle phrase means the following:
"But because a door of communication between that and the Doctor's study, where there was alight, was open,"
Thank you in advance for your help.