  Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split
category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the
multiplicative group of $k$, and consider the resulting twisted category
algebra $A:=k_\alpha\mathsf{C}$. Several interesting algebras arise that way,
for instance, the Brauer algebra. Moreover, the category of biset functors over
$k$ is equivalent to a module category over a condensed algebra $\varepsilon
A\varepsilon$, for an idempotent $\varepsilon$ of $A$. In [2] the authors
proved that $A$ is quasi-hereditary (with respect to an explicit partial order
$\le$ on the set of irreducible modules), and standard modules were given
explicitly. Here, we improve the partial order $\le$ by introducing a coarser
order $\unlhd$ leading to the same results on $A$, but which allows to pass the
quasi-heredity result to the condensed algebra $\varepsilon A\varepsilon$
describing biset functors, thereby giving a different proof of a quasi-heredity
result of Webb, see [26]. The new partial order $\unlhd$ has not been
considered before, even in the special cases, and we evaluate it explicitly for
the case of biset functors and the Brauer algebra. It also puts further
restrictions on the possible composition factors of standard modules.
