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. A very basic example is the following: Define a (class) well ordering on $\text{On} \times \text{On}$ by

$(\alpha,\beta) < (\gamma,\delta) \Leftrightarrow \max(\alpha,\beta) < \max(\gamma,\delta) \vee (\max(\alpha,\beta) = \max(\gamma,\delta) \wedge$ $(\alpha < \gamma \vee (\alpha = \gamma \wedge \beta < \delta))$.

Its type can be used to define a bijection of classes $\text{On} \cong \text{On} \times \text{On}$, but also it yields the equality $\kappa^2=\kappa$ for every cardinal number $\kappa \geq \aleph_0$ (even without AC).

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. A very basic example is the following: Define a (class) well ordering on $\text{On} \times \text{On}$ by

$(\alpha,\beta) < (\gamma,\delta) \Leftrightarrow \max(\alpha,\beta) < \max(\gamma,\delta) \vee (\max(\alpha,\beta) = \max(\gamma,\delta) \wedge$ $(\alpha < \gamma \vee (\alpha = \gamma \wedge \beta < \delta))$.

Its type can be used to define a bijection of classes $\text{On} \cong \text{On} \times \text{On}$, but also it yields the equality $\kappa^2=\kappa$ for every cardinal number $\kappa \geq \aleph_0$ (even without AC).

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 $\text{On} \times \text{On}$.

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. A very basic example is the following: Define a (class) well ordering on $\text{On} \times \text{On}$ by

$(\alpha,\beta) < (\gamma,\delta) \Leftrightarrow \max(\alpha,\beta) < \max(\gamma,\delta) \vee (\max(\alpha,\beta) = \max(\gamma,\delta) \wedge$ $(\alpha < \gamma \vee (\alpha = \gamma \wedge \beta < \delta))$.

Its type can be used to define a bijection of classes $\text{On} \cong \text{On} \times \text{On}$, but also it yields the equality $\kappa^2=\kappa$ for every cardinal number $\kappa \geq \aleph_0$ (even without AC).