Left residual operator for composition

Property-theoretic statement
Suppose $$p,q$$ are two subgroup properties. The left residual of $$p$$ by $$q$$ is the unique subgroup property $$r$$ such that:

$$s * q \le p \iff s \le r$$.

Here, $$*$$ denotes the composition operator.

Statement with symbols
Suppose $$p,q$$ are two subgroup properties. The left residual of $$p$$ by $$q$$ is defined as the subgroup property $$r$$ as follows:

$$H$$ has property $$r$$ in $$G$$ if whenever $$G$$ is embedded as a subgroup with property $$q$$ in a group $$K$$, $$H$$ has property $$p$$ in $$K$$.

Relation with left transiter
The left transiter of a subgroup property $$p$$ is defined as the left residual of $$p$$ by itself.