by $p {\mathrel{\oplus}}{\{s\}}\, p' = (p + p') \mod (s + 1)$, satisfying all the required laws.

$$\xymatrix@C=4.8em@R=1.2em{
& & & & & \\
& & & & & \\
\bullet \ar@{->}@/_1pc/[rr]_(0.8){p=2} \ar@{->}@/^1pc/[rrrr]^{p {\mathrel{\oplus}}p' = 2 + \, 2 = 4} \ar@{-}[uurr]_>{s = {{\ast}}} \ar@{-}[ddrr] & \bullet  & \bullet \ar@{.>}@/_1pc/[rr]_(0.8){p' = 2} \ar@{.}[uurr]_>{s' = s {\mathrel{\downarrow}}p = {{\ast}}} \ar@{.}[ddrr] & \bullet & \bullet & \bullet & ... \\
& & & & & \\
& & & & &
}$$

The container of *streams* ($S = 1$, $P\, {{\ast}}= {\mathsf{Nat}}$) carries a very trivial directed container structure given by ${{\ast}}{\mathrel{\downarrow}}p =
  {{\ast}}$, ${\mathsf{o}}= 0$ and $p {\mathrel{\oplus}}p' = p +
  p'$. Fig. \[fig:dcontainerstream\] shows how a position $p = 2$ in the only possible global shape $s = {{\ast}}$ and a position $p' = 2$ in the equal subshape $s' = s {\mathrel{\downarrow}}p = {{\ast}}$ give back a position $p + p'
  = 4$ in the global shape.

This directed container is nothing else than the monoid $({\mathsf{Nat}},
  0, +)$ seen as a directed container.

Similarly to the theory of containers, one can also define morphisms between directed containers.

A *morphism* between two directed containers $(S \lhd P, {\mathrel{\downarrow}}, {\mathsf{o}}, 