Endomorphing form

Definition
Let $$V$$ be a variety of algebras. An endomorphing form for $$V$$ is an algebraic expression in $$k+1$$ variables, of which one variable is the input variable and the other $$k$$ variables are parameters, such that:

For any algebra $$A \in V$$ and any choice of values for the $$k$$ parameters, the map sending the variable to the value of the expression, is an endomorphism of the algebra.

An automorphism of an algebra that arises from an automorphing form with respect to the variety is termed a form-endomorphism.