Dear Prof. Kuhn, thank you for your counterexample. I had this suspicion since, under the aforementioned conditions and according to the proof, any map of the form $X\xrightarrow{\iota_1} BP_u\xrightarrow{Bi_u} BG\xrightarrow{tr} BP_v\xrightarrow{\pi} X$ can be seen of the form $X\xrightarrow{\iota_2} BP_v\xrightarrow{Bi_v} BG\xrightarrow{tr} BP_v\xrightarrow{\pi} X$ modulo an ideal $I_{uv}$. Maybe this ideal makes it possible, but I do not know how, any suggestion? please.

@Jeff Can you please explain why $[A,X]\xrightarrow{\sigma_{\star}}[A,\Omega\Sigma X]$ is a bijection? (this map is supposed to be an isomorphism according to ncatlab).

@Fernando I apologize, I meant the coequalizer when the commutative ring is a field.

@Fernando is it also complicated to construct equalizers when the commutative ring is a field?.

I see, this assertion is proposition 1.8 in the article by Getzler and Goerss and is proved by using a strange a strange category to me named the category of **profinite ** differential graded algebras, can you please tell me a more direct argument to deduce such a result?.

Thank you for the clarification. I apologize for my delay in aswering, I tried to better understand your edit. With this result in hand, is it reasonable to conclude that the category of dg coalgebras has all (small) limits?. I mean, by applying dualization on the finite dimensional dg subcoalgebras and assuming the existence of all small colimits in the category of dg algebras.

Thank you!. Apparently you did not need $C$ to be counital, did you?.