Full invariance is transitive for any variety of algebras

Statement
Suppose $$\mathcal{V}$$ is a variety of algebras. Suppose $$A$$ is an algebra of $$\mathcal{V}$$ and $$B$$ is a fully invariant subalgebra of $$A$$ (in other words, every endomorphism of $$A$$ sends $$B$$ to itself). Suppose $$C$$ is a fully invariant subalgebra of $$B$$. Then, $$C$$ is a fully invariant subalgebra of $$A$$.

Particular cases

 * Full invariance is transitive
 * Full invariance is transitive for Lie rings

Other similar facts

 * Characteristicity is transitive for any variety of algebras
 * Characteristicity is transitive
 * Characteristicity is transitive for Lie rings
 * Strict characteristicity is transitive for any variety of algebras
 * Derivation-invariance is transitive for any subvariety of the variety of rings