Consider the oriented n-dimensional hypercubes $C_n$.C_n$.

And so on, inductively over $n \in \mathbb{N}$. mathbb{N}$. Some of the objects (orthey can be $0$.0$.

Example 1: $I = { \xymatrix{X \ar@<+0.35ex>[r]^f { X \ar@<-0.35ex>[r]_g & overset{f}{\underset{g}\rightrightarrows} Y } }$ \}$ can bearranged on $C_2$:
C_2$:
$$ \xymatrix{ X \ar[r]^f & \ar[d]_g overset{f}\rightarrow & Y \ar[d] \ {\scriptstyle g}\downarrow && \downarrow \\ Y & \ar[r] rightarrow & 0.\end{center}end{array} $$

Consider the oriented n-dimensional hypercubes $C_n$.
$C_0$: one object $X_0$.
$C_1$: $X_0 \to X_1$.
$C_2$:
\begin{center} \xymatrix{ X_{00} \ar[r] \ar[d] & X_{01} \ar[d] \ X_{10} \ar[r] & X_{11} . } \end{center}
$C_3$:
\begin{center} \xymatrix{ X_{000} \ar[rr] \ar[dd] \ar[dr] && X_{010} \ar@{-->}[dd] \ar[dr] \ & X_{100} \ar[rr] \ar[dd] && X_{110} \ar[dd] \ X_{001} \ar@{-->}[rr] \ar[dr] && X_{011} \ar@{-->}[dr] \ & X_{101} \ar[rr] && X_{111} . } \end{center}
And so on, inductively over $n \in \mathbb{N}$. Some of the objects (or everyone) filling the vertices can eventually be the same. Moreover, they can be $0$.

Let $I$ be an arbitrary finite oriented diagram (graph) with no cycles (composable arrows starting and ending on one single object).

My question is: Does always exist an $n \in \mathbb{N}$ such that $I$ is included in $C_n$? (References are also welcomed.)

Example 1:
$I = { \xymatrix{X \ar@<+0.35ex>[r]^f \ar@<-0.35ex>[r]_g & Y} }$ can be arranged on $C_2$:
\begin{center} \xymatrix{ X \ar[r]^f \ar[d]_g & Y \ar[d] \ Y \ar[r] & 0 . } \end{center}