There is actually a precise moment when the policeman, who is Jimmy Wells, realizes that the man standing in the darkened doorway of the hardware store is the man who is wanted by the Chicago police. But the reader does not also realize this at the same time. In fact the reader does not understand this until the end of the story.
Jimmy Wells knows that the man in the doorway is his old friend Bob as soon as he confronts him. This is because Jimmy made a date to meet Bob at that location, which used to be a restaurant, twenty years ago when they parted. Bob may have changed in twenty years, but Jimmy knows it is his old friend in spite of the darkness because Bob immediately starts talking.
“It's all right, officer,” he said, reassuringly. “I'm just waiting for a friend. It's an appointment made twenty years ago.
Bob continues talking about the past. Obviously he knows so much about Jimmy Wells that he must be Bob. But then the confident Bob lights his cigar. This is when Jimmy is able to see his face for the first time. And this is when he realizes that his old friend Bob is also the man wanted by the Chicago police. But the reader does not realize this fact until the end, when the arresting plainclothes officer hands 'Silky' Bob a note from Jimmy Wells, who was the uniformed cop Bob was talking to without realizing it was his old pal Jimmy. The note reads as follows:
Bob: I was at the appointed place on time. When you struck the match to light your cigar I saw it was the face of the man wanted in Chicago. Somehow I couldn't do it myself, so I went around and got a plainclothesman to do the job. JIMMY.
It should be noted that in those days, around 1906, there was no way of sending photographs or even sketches by telegraph wire. What probably happened was that the Chicago police sent the New York police a wire with a description of 'Silky' Bob's age and general appearance, especially his face. When 'Silky' Bob lights his cigar, this is what Jimmy sees:
The man in the doorway struck a match and lit his cigar. The light showed a pale, square-jawed face with keen eyes, and a little white scar near his right eyebrow. His scarf pin was a large diamond, oddly set.
Jimmy's old friend fits the general description of the man called 'Silky' Bob who is wanted by the Chicago police. In addition there are two facts that provide nearly positive identification. The man in the doorway has a white scar near his right eyebrow, and he is wearing an "oddly set" diamond scarf pin. Both these details would have been included in the telegram from Chicago. In fact, the scarf pin would have been more precisely described. For example, it might have been a large diamond surrounded by thirteen small diamonds or rubies representing the original thirteen American colonies, or something comparably unique.
When the plainclothes officer arrives on the scene about twenty minutes later, he still makes sure he is arresting the wanted man.
The two men started up the street, arm in arm. The man from the West, his egotism enlarged by success, was beginning to outline the history of his career.The other, submerged in his overcoat, listened with interest.
Thinking he is talking to his old friend Jimmy Wells, Bob gives the detective all the information he needs to be sure he is arresting the right man.