Is there a multiplication on the tensor product of an algebra and a coalgebra?

Question

I am reading in Algebraic Operads (Vallette & Loday). In the subchapter dealing with the convolutionproduct, they write

"Any linear map $\alpha:C\to A$ from a graded coalgebra $C$ to a graded algebra $A$, defines a morphism $$C\to_{\Delta} C\otimes C\to_{Id\otimes \alpha}C\otimes A$$ which induces a unique derivation on $C\otimes A$.."

If I have understood things right a derivation should satisfy Leibniz rule so there should be a multiplication on $C\otimes A$, how do we define this multiplication?

Answer 1

The answer was on the page before if anybody was wondering.... They mean an $A$-derivation of $C\otimes A$ as an $A$-module where $(A,d_A)$ is a dga-algebra.

An $A$-derivation of a right $A$-module $M$ is a linear map $$d:M\to M$$ that satisfies $$d_M(ma)=d_M(m)a\pm md_A(a)$$