  Prestacks are algebro-geometric objects whose defining relations are far from
quadratic. Indeed, they are cubic and quartic, and moreover inhomogeneous. We
show that box operads, a rectangular type of operads introduced in
arXiv:2305.20036, constitute the correct framework to encode them and resolve
their relations up to homotopy. Our first main result is a Koszul duality
theory for box operads, extending the duality for (nonsymmetric) operads. In
this new theory, the classical restriction of being quadratic is replaced by
the notion of being $\textit{thin-quadratic}$, a condition referring to a
particular class of ``thin'' operations. Our main case of interest is the box
operad $Lax$ encoding lax prestacks as algebras. We show that $Lax$ is not
Koszul by explicitly computing its Koszul complex. We then go on to remedy the
situation by suitably restricting the Koszul dual box cooperad
$Lax^{\antishriek}$ to obtain our second main result: we establish a minimal
(in particular cofibrant) model $Lax_{\infty}$ for the box operad $Lax$
encoding lax prestacks. This sheds new light on Markl's question on the
existence of a cofibrant model for the operad encoding presheaves of algebras
from arXiv:math/0103052. Indeed, we answer the parallel question with
presheaves viewed as prestacks, and prestacks considered as algebras over a box
operad, in the positive.
