The article Limits of Coalgebras, Bialgebras and Hopf Algebras offers two descriptions for the equalizer of two unital coassociative coalgebras over a field. The latter description (Remark 1.2) is explicit and claims that, given $f,g:C\rightarrow D$ two morphisms of coalgebras, the equalizer $(E,i)$ is given by
$$E=\lbrace{ c\in C/ c_{(1)}\otimes f(c_{(2)})\otimes c_{(3)}=c_{(1)}\otimes g(c_{(2)})\otimes c_{(3)}}\rbrace$$
and $i:E\rightarrow C$ the inclusion.
I understand this is indeed the equalizer when $E,D$ are Hopf algebras, by using the antipode. However, I cannot conclude the same for coalgebras. Any suggestion?, please. Is there a similar description for dg coalgebras?. Thanks.