$\mathsf{Bij}(\kappa) := \{f: X \rightarrow Y \mid |X| = \kappa \mbox{ and } f : X \rightarrow Y \mbox{ is a bijection}\}$That is, $\mathsf{Bij}(\kappa)$ is the class of bijections between sets of cardinality $\kappa$.

$\mathsf{Bij}(\kappa)$ is a groupoid, and therefore a category, such that :

i. the objects of $\mathsf{Bij}(\kappa)$ are sets $X$ of cardinality $\kappa$;where, for any object $X$, for any $x \in X$, $i(x) = x$.

ii. the morphisms of $\mathsf{Bij}(\kappa)$ are the bijections $X \rightarrow Y$.

iii. for each $X$, the identity morphism $1_X$ is $i: X \rightarrow X$.

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