I'm uncertain about the first answer I came up with:

\\( 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...

\\( 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...