Given a function $f: A \times B \rightarrow C$, I'd like to define a special case where $A$ could be the empty set. In this case, I would like the function to "decay" into $f: B \rightarrow C$, i.e. simply ignore the (non-existent) $A$ value. Is there a canonical way of doing this?
This seems related to the empty function and How to construct a function from a pair of possibly empty sets?, but I'm not sure how this would be applied to a function taking two inputs.