directed container morphisms for the directed container of non-empty lists and suffixes (and the identity directed container). But those of self-append and reversal are not.

For the directed container of non-empty lists and cyclic shifts, not only the representations of the head and drop-even functions but also the self-append function are directed container morphisms.

The identities and composition of ${\mathbf{Cont}}$ can give the identities and composition for directed containers, since for every directed container $E = (C, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$, the identity container morphism ${\mathsf{id}^\mathrm{c}}\, {\{C\}}$ is a directed container morphism and the composition $h {\mathrel{\circ^\mathrm{c}}}h'$ of two directed container morphisms is also a directed container morphism.

Directed containers form a category ${\mathbf{DCont}}$.

Interpretation of Directed Containers
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As directed containers are containers with some operations obeying some laws, a directed container should denote not just a set functor, but a set functor with operations obeying some laws. The correct domain of denotation for directed containers is provided by comonads on sets.

Given a directed container $E = (S \lhd P, {\mathrel{\downarrow}}, {\mathsf{o}}, {\mathrel{\oplus}})$, we define its *interpretation* ${\llbracket E \rrbracket^\mathrm{dc}}$ to be the set functor $D =
{\llbracket S \lhd P \rrbracket^\mathrm{c}}$ (i.e., the interpretation of the underlying container) 