\$$X = { a b c } \$$

\$$Y = { ● ▲ ☐ } \$$

\$$f(a) = ▲ \$$

\$$f(b) = ▲ \text{or} ☐ \$$

\$$f(c) = ● \$$

then

\$$P := { { ▲ ☐ } { ● } } \$$

\$$f*({▲ ◻︎}) = { a b } \$$

\$$f*({ ● }) = { c } \$$

and

\$$Q = { { ▲ } { ☐ } { ● } } \$$

\$$f*({ ▲ }) = { a } \$$

\$$f*({ ☐ }) = { b } \$$

\$$f*({ ● }) = { c } \$$

Beyond the construction of the examples above, I'm tempted to think of f* as a function, but that feels incorrect and I can't quite say why...