 On the syntactic side, the free \(C\)-algebra \(B\) on a set \(X\) arises as a quotient of the term algebra formed from \(X\) (viewed as a set of variables) using the operation symbols and constants common to the algebras of \(C\). The quotient identifies those terms that have the same value for all algebras \(A\) of \(C\) and all valuations assigning values in \(A\) to the variables of \(X\). This performs just enough identifications to satisfy every law of \(C\) (thereby making this quotient a \(C\)-algebra) while still retaining the syntactic essence of the original term algebra in a sense made more precise by the following paragraph.