# Categorical product of non-unital associative differential graded coalgebras

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.