also introduce strict directed containers and construct the product of two strict directed containers in the category of directed containers. Intuitively, a strict directed container is a directed container where no position in a non-root subshape of a shape translates to its root. In Section \[sec:monads\], we ask whether a similar characterization is possible for containers that are monads and hint that this is the case. In Section \[sec:cointerp\], we show that interpreting the opposite of the category of directed containers into set functors gives monads. In Section \[sec:polycom\], we hint how the directed container theory (presented in the paper for ${\mathbf{Set}}$) could be developed in the more general setting of categories with pullbacks. We briefly summarize related work in Section \[sec:related\] and conclude with outlining some directions for future work in Section \[sec:concl\]. The proofs of the main results of Sections \[sec:dcontainers\] and \[sec:constructions\] appear in Appendices A and B.

We spend a section on the background theory of containers as they are central for our paper but relatively little known, but assume that the reader knows about comonads, monoidal categories and comonoids.

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This article is a revised and 