MATH.5:4.3 - Establish preservation and uniqueness

For freely formed expressions, preservation follows from the defining clause: evaluating an operation on expressions gives the target operation on their evaluations. The generator and constant clauses cover the starting cases.

Suppose another operation-preserving map H has the same assigned generator values. It agrees with E on generators and constants. If it agrees on the constituents of an expression, preservation forces agreement on the whole expression. Induction therefore gives H(t)=E(t) for every expression.

The conclusion is uniqueness among maps preserving the named operations and agreeing with the given assignment. Changing the assignment or required operations changes that question.

For words, the corresponding argument starts with the empty word and extends by one generator. Every concatenation-preserving map with the same identity and generator images must return the same ordered product.

For paths, the base case uses the identity at each F(X). Induction on the second path’s length, using target associativity, gives E(p;q)=E(p) star E(q) for every permitted join. A map preserving identities and composition with the same object and generator assignments must follow those recursive clauses, so it is unique. Such an object-and-arrow map between categories is called a functor. The one-object word construction is its monoid case.