What buildings did Holmes and Watson see on the back side of the block where the pawnbroker's shop was in "The Red-Headed League"?
They find a bank behind the pawnbroker’s shop.
When a pawnbroker comes to Sherlock Holmes with a suspicious story of a job at a Red-Headed League that no longer exists, Holmes is curious. He asks some questions and realizes that there is something going on. One visit to the shop confirms it.
The neighborhood of the pawnbroker’s is Saxe-Coburg Square, which is a “poky, little, shabby-genteel place.” Holmes pretends to be asking directions so that he can get his eyes on the man he thinks is John Clay. His suspicions are confirmed. He tells Watson that the man is one of the most daring men in England, and also one of the smartest.
Holmes tells Watson they are in enemy territory, and they have a look around. He prides himself on knowing the city well, and wants to confirm that he knows what buildings are there.
They walk around behind the shop, to see what is there.
It is a hobby of mine to have an exact knowledge of London. There is Mortimer's, the tobacconist; the little newspaper shop, the Coburg branch of the City and Suburban Bank, the Vegetarian Restaurant, and McFarlane's carriage-building depot. That carries us right on to the other block.
Once they have had a look around, Holmes decides it’s time for some food and violin. Holmes spends a peaceful and quiet afternoon. It is clear that the most interesting building on that block was the bank. Holmes has confirmed his suspicions yet again.
Holmes is aware that an infamous criminal is impersonating a pawnbroker’s assistant. He also knows that the pawn shop is next to a bank. It is logical to assume that the “assistant” is going to rob the bank. Holmes arranges to catch him in the act, because it’s much better to stop an attempted bank robbery while it’s happening than attempt to arrest someone for planning one.