I'm reading articles about applications of Hopf algebras in physics. In one of them there is following definition of graded coalgebra:

A coalgebra $C$ is called graded if $C=\bigoplus\limits_{n\ge 0}C(n)$ as vector space and $\forall n\ge 0 : \ \Delta (C(n))\subset \bigoplus_\limits{i=0}^{n}C(i)\otimes C(j)$ and $\forall n\ge 1 : \varepsilon|_{C(n)}=0$

where $\Delta$ is a coproduct and $\varepsilon$ is a counit.

In the other article one can find definition in which it is assumed only that

$\forall n\ge 0 : \ \Delta (C(n))\subset \sum_\limits{i=0}^{n}C(i)\otimes C(j) $

Are these definitions equivalent ?

I'm reading articles about applications of Hopf algebras in physics. In one of them there is following definition of graded coalgebra:

A coalgebra $C$ is called graded if $C=\bigoplus\limits_{n\ge 0}C(n)$ as vector space and $\forall n\ge 0 : \ \Delta (C(n))\subset \bigoplus_\limits{i=0}^{n}C(i)\otimes C(n-i)$ and $\forall n\ge 1 : \varepsilon|_{C(n)}=0$

where $\Delta$ is a coproduct and $\varepsilon$ is a counit.

In the other article one can find definition in which it is assumed only that

$\forall n\ge 0 : \ \Delta (C(n))\subset \sum_\limits{i=0}^{n}C(i)\otimes C(n-i) $

Are these definitions equivalent ?