compouned (composite) unique keys
Every table should have a PRIMARY KEY.  In MySQL this implies two things: a UNIQUEness constraint, and an index.
Any kind of index can be "composite" ("compound"), that is be composed of more than one column.
It is rarely useful to have two UNIQUE keys in a single table.  (Remember: PRIMARY counts as UNIQUE.
A common exception is when you are 'normalizing' a long string and 'mapping' it to a short INT.
If you already have a unique key (a), it is unreasonable and unnecessary to have also have UNIQUE(a,b) or UNIQUE(b,a).  On the other hand, it may be useful to have a non-unique INDEX(a,b) and/or INDEX(b,a).  Think how the uniqueness of (a) implies the uniqueness of the other two.
With INDEX(a,b), there is virtually no reason to also have INDEX(a)  (Note:  I am not talking about UNIQUE in this bullet item.)


If I have not addressed your specific question (perhaps indirectly), please rephrase your question.