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How do I use a binary operator (such as union or direct sum) as a large operator (such as sum or product)?

First of all, you need to define the desired symbol as a personal user symbol, as describe in the FAQ How do I use a symbol not provided in Math's Catalog of characters?.

For instance, if you named union symbol %union, you should enter:

oper %union from i in I A_i

to get the following formula: \underset{i\in I}{U}{A}_{i}

All symbols defined by user can be used as large operator if they are preceded with oper, like in previous example.