Modules in combinatorics

Consider (the totally disconnected groupoid $\boxed\Perm$ of permutations of sets),
which (considered as a category) is a subcategory of (the category $\Set$): $\quad \Perm \subset \Set$.
Recall that any category, say $\calC$, is a left-right bimodule over itself, via its hom bifunctor: $\smash{ \quad \calC\op \times \calC \xrightarrow{\textstyle \hom - \calC -} \Set }\;$.
In particular (the category $\Set$) is a left-right bimodule over itself: $(\functionf,\functiong,\functionh) \mapsto \functionf\functiong\functionh$.
By restriction of scalars, $\Set$ is a bimodule over (its groupoid subcategory $\Perm$): $\smash{ \quad \Perm\op\times\Perm \xrightarrow{\textstyle \hom - \Set -} \Set \; : \; (\sigma, \tau) \mapsto ( \hom \sigma \Set \tau \; : \; \functionf \mapsto \sigma \functionf \tau ) } \;$.
(Analogy: In algebra, any ring is a left-right bimodule (Wikipedia, nLab) over each of its subrings.)

Take the left, right, and left-right quotients of $\Set$ by $\Perm$.
This yields [(the first four parts) of (the "twelvefold way")] .

Now consider the subcategories of surjections, $\boxed\Surj$, and, of injections, $\boxed\Inj$, of the category $\Set$.
Each is closed under both left and right actions of $\Perm$, thus is a sub-$\Perm$-bimodule.
Applying the same (quotient by $\Perm$) constructions to (those two submodules) gives [(the other eight parts) of (the twelvefold way)].

We can present the twelvefold way as a table:

The Twelvefold Way

Module, with scalars	Module, without scalars	Left Quotient indistinguishable balls	Right Quotient indistinguishable boxes	Left-Right Quotient indistinguishable balls and boxes
$(\Perm, \Set)$	$\Set$	$\Perm\backslash\Set$	$\Set/\Perm$	$\Perm\backslash\Set/\Perm$
$(\Perm, \Surj)$	$\Surj$	$\Perm\backslash\Surj$	$\Surj/\Perm$	$\Perm\backslash\Surj/\Perm$
$(\Perm, \Inj)$	$\Inj$	$\Perm\backslash\Inj$	$\Inj/\Perm$	$\Perm\backslash\Inj/\Perm$

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Consider the intersection of the submodules $\Surj$ and $\Inj$, the groupoid $\smash{\boxed\Bij}$ of bijections of sets
(which of course contains $\Perm$ as a subgroupoid: $\Perm \subset \Bij \subset \Surj,\Inj \subset \Set$).
Given any three equipotent sets, say $\setX$, $\setY$, and $\setZ$,
consider commutative triangles in $\Bij$ with those fixed sets as vertices.
Fixing one side of such a triangle to a specific (internal) bijection
sets up (an external bijection between the possible fill-ins for the other two sides).
Thus, for example, choosing (an internal bijection from $\setX$ to $\setY$)
sets up (an external bijection from $\hom \setY \Set \setZ$ to $\hom \setX \Set \setZ$),
which restricts to (an external bijection from $\hom \setY \Bij \setZ$ to $\hom \setX \Bij \setZ$).

If we have an arbitrary finite set $\setY$, of size $n$,
let $\setX = \mathbf n$, the canonical linear order of size $n$.
Then

choosing a particular linear ordering of $\setY$ (a specific element of $\hom {\mathbf n} \Bij \setY$) sets up
an external bijection from {permutations of $\setY$ ($\hom \setY \Perm \setY = \hom \setY \Bij \setY$)} to {linear orders on $\setY$ ($\hom {\mathbf n} \Bij \setY$)},

a result which is frequently used, e.g., rather extensively in Stanley, Enumerative Combinatorics.

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References

[CKVW] Carboni, A.; Kelly, G. M.; Verity, D.; Wood, R. J. (1998). "A 2-Categorical Approach To Change Of Base And Geometric Morphisms II". Theory and Applications of Categories. 4 (5): 82–136.