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Given two non-unital associative dg coalgebras $D$ and $C$, I want to give an explicit construction of the product $C\prod D$, this may follow from the dual construction (coproduct of non-unital associative dg algebras). I have read the article A closed model structure for differential graded algebras by Jardine and found that given two unital associative dg algebras $A,B$, we have: $$A\coprod B=T(A\otimes B)/I$$ where $I$ is the ideal generated by elements of the form $(a_1\otimes b_1)\otimes (1\otimes b_2)-(a_1\otimes b_1b_2)$ and $(a_1\otimes 1)\otimes (a_2\otimes b_2)-(a_1a_2\otimes b_2)$.

How could I reformulate this quotient when both $A$ and $B$ have no units?.

In the case of coalgebras, I would appreciate a few hints to make the dual construction. Thanks.

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