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If $X,Y$ are classes defined by formulas $\phi(x), \psi(y)$, then a map $X \to Y$ is just a formula $\alpha(x,y)$ such that $\forall x (\phi(x) \Rightarrow \exists^1 y (\psi(y) \wedge \alpha(x,y)))$. Here $\exists^1$ abbreviates "there exists exactly one ...". This defines the (meta)category of classes and maps of classes. The isomorphisms are exactly the bijections, i.e. with the above notation the maps $\alpha : X \to Y$ such that $\forall y (\psi(y) \Rightarrow \exists^1 x (\phi(x) \wedge \alpha(x,y)))$. In this MO thread it was shown that Schröder Bernstein holds in this setting.
I expect that you can find this notion of bijection in almost every introduction to set theory. For example, A very important for some basic constructions example is the bijection following: Define a (class) well ordering on $\text{On} \cong \text{On} \times \text{On}$ which is given bythe order
$(\alpha,\beta) < (\gamma,\delta) \Leftrightarrow \max(\alpha,\beta) < \max(\gamma,\delta) \vee (\max(\alpha,\beta) = \max(\gamma,\delta) \wedge$ $(\alpha < \gamma \vee (\alpha = \gamma \wedge \beta < \delta))$.
Its type of can be used to define a certain class well-ordering on bijection of classes $\text{On} \cong \text{On} \times \text{On}$.text{On}$, but also it yields the equality$\kappa^2=\kappa$for every cardinal number$\kappa \geq \aleph_0$(even without AC). 1 If$X,Y$are classes defined by formulas$\phi(x), \psi(y)$, then a map$X \to Y$is just a formula$\alpha(x,y)$such that$\forall x (\phi(x) \Rightarrow \exists^1 y (\psi(y) \wedge \alpha(x,y)))$. Here$\exists^1$abbreviates "there exists exactly one ...". This defines the (meta)category of classes and maps of classes. The isomorphisms are exactly the bijections, i.e. with the above notation the maps$\alpha : X \to Y$such that$\forall y (\psi(y) \Rightarrow \exists^1 x (\phi(x) \wedge \alpha(x,y)))$. In this MO thread it was shown that Schröder Bernstein holds in this setting. I expect that you can find this notion of bijection in almost every introduction to set theory. For example, very important for some basic constructions is the bijection$\text{On} \cong \text{On} \times \text{On}$which is given by the order type of a certain class well-ordering on$\text{On} \times \text{On}\$.