Characteristicity is transitive for any variety of algebras

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ANALOGY: This is an analogue in variety of algebrass of a fact encountered in group. The old fact is: characteristicity is transitive.
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Statement

Suppose \mathcal{V} is a variety of algebras. Suppose A is an algebra of \mathcal{V}, B is a subalgebra of A that is characteristic in A (i.e., every automorphism of A sends B to itself) and C is a subalgebra of B that is characteristic in B. Then, C is a characteristic subalgebra of A.

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Particular cases

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Proof

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