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The composition of two morphisms of coalgebras is again a morphism of coalgebras.

Is the corresponding statement true for algebras? i.e. Is the composition of two morphisms of algebras is again a morphism of algebras?

Sincere thanks for any help.

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Yes. But your question is not absolutely clear to me. There are two different notions of algebra that could be relevant here, even though I believe you are talking about the first:

  1. A set together with function, relations, and constants.

  2. A vector space with a multiplication that interacts nicely with the vector space structure.

The usual notion of morphism in both cases would be homomorphism, but you have to make sure that your algebras are of the same signature, i.e., in the case of 1., both algebras have the same number of $n$-ary function, $n$-ary relations and constants, because otherwise you don't have a notion of homomorphism between your structures.

It is easily checked that composition of homomorphisms are again homomorphisms. The word "morphism" is usually used in category theory, and in category theory the composition of two morphisms is a morphism by the definition of a category.