expanded version of the FoSSaCS 2012 conference paper [@ACU]. We have added many of the proofs that were omitted from the conference version. We have rearranged the different constructions on directed containers into a separate section, namely Section \[sec:constructions\]. In Section \[sec:cofree\], we give a detailed discussion of cofree directed containers. In Section \[sec:products\], which is entirely new, we define strict directed containers and coideal comonads and give an explicit formula for the product of two strict directed containers.

Likewise entirely new are the sections on cointerpreting directed containers in Section \[sec:cointerp\] and directed containers in categories with pullbacks in Section \[sec:polycom\].

Containers {#sec:containers}
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We begin with a recap of containers. We introduce the category of containers and the fully-faithful functor into the category of set functors defining the interpretation of containers and show that these are monoidal. We also recall some basic constructions of containers. For proofs of the propositions in this section and further information, we refer the reader to Abbott et al. [@Abbott2005; @abbott:phd].

Containers {#containers}
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Containers are a form of “syntax” for datatypes. A *container* $S \lhd P$ is given by a set $S : {\mathsf{Set}}$ of *shapes* and a shape-indexed family $P : 