Subhomomorphism relation between transfer condition operator and composition operator

Statement
Let $$T$$ denote the fact about::transfer condition operator on subgroup properties: given a subgroup property $$p$$, the property $$T(p)$$ is the weakest property satisfying the transfer condition, that is stronger than $$p$$.

Let $$*$$ denote the fact about::composition operator.

Then:

$$T(p) * T(q) \le T(p * q)$$.

Similar facts

 * Subhomomorphism relation between transfer-closure operator and intermediately operator over composition operator

Corollaries

 * Transfer condition is composition-closed
 * Transitivity is transfer condition operator-preserved