  Game comonads, introduced by Abramsky, Dawar and Wang and developed by
Abramsky and Shah, give an interesting categorical semantics to some
Spoiler-Duplicator games that are common in finite model theory. In particular
they expose connections between one-sided and two-sided games, and parameters
such as treewidth and treedepth and corresponding notions of decomposition. In
the present paper, we expand the realm of game comonads to logics with
generalised quantifiers. In particular, we introduce a comonad graded by two
parameters $n \leq k$ such that isomorphisms in the resulting Kleisli category
are exactly Duplicator winning strategies in Hella's $n$-bijection game with
$k$ pebbles. We define a one-sided version of this game which allows us to
provide a categorical semantics for a number of logics with generalised
quantifiers. We also give a novel notion of tree decomposition that emerges
from the construction.
