What conditions must be checked for that $c$ is Cohen over $V$. $\textbf{Hechler forcing} $
Let $\mathbb{D}=\{(s,f): s \in \omega^{<\omega},f\in \omega^{\omega} \text{and} s \subseteq f\}$, ordered by $(t,g)\leq (s,f)$ if $s \subseteq t$, $g$ dominates $f$ everywhere, and $f(i) \leq t(i)$ for all $i \in |t|\setminus|s|$. It generically adds a new real $d=\bigcup\{s:(s,f)\in G$for some $f \in \omega^{\omega}\}$.
I want to show that $\mathcal{Hechler forcing}$ adds a $\mathcal{Cohen}$  real.
Let $d \in \omega^{\omega}$ be a Hechler  real over $V$. Define $c \in 2^\omega$ by $c(n)=d(n)$ mod $2$.
What conditions must be checked for that $c$ is Cohen over $V$.
$\textbf{Eventually different forcing}$ $\mathbb{E}$ consists of pairs $(s,F)$, , where $s \in \omega^{<\omega}$ and $F$ is a finite set of reals with
$(s,F)\leq (t,G)$ iff $t \subseteq s$ and $G \subseteq F$ and $\forall{i \in \text{dom}(s\setminus t)}\forall{g \in G}(s(i)\neq g(i))$.It generically adds a new real $f_{G}=\bigcup\{s:(s,H)\in G\}$. 
I want to show that Eventually different forcing adds a $\mathcal{Cohen}$  real.
If $f_{G} \in \omega^{\omega}$ is a generic real, Define $c \in 2^\omega$ by $c(n)=1$ if $f_{G}$ is even or
$c(n)=0$ if $f_{G}$ is odd
What conditions must be checked for that $c$ is Cohen over $V$.
Any suggestion please.

A real $c\in2^\omega$ is Cohen over $V$ if and only if, for all $D\subseteq2^{<\omega}$, if $D$ is dense in $2^{<\omega}$ (meaning that every $s\in2^{<\omega}$ has an extension in $D$) and $D\in V$, then $D$ contains an initial segment $c\upharpoonright n$ of $c$.
