A subway is an underground railway system used to transport large numbers of passengers within urban and suburban areas. Subways are usually built under city streets for ease of construction, but they may take shortcuts and sometimes must pass under rivers. Link
The use of a subway system as a means of transportation for modern train has made the whole idea of a subway system all about trains. But the use of under ground water pipes below houses, especially where there meet was called a subway too, it still is though.
Even the passengers moving from one place to another in the subway and some waiting in for their transit is made to know that the system was made for people to move from one place to another.
So down the subway is the usual going down the subway to conduct one business or the other.
Most times it is very good to have an idea of a word and what it means, even when it can have different meaning.
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