The Quotient of a Set by an Equivalence Relation and the Canonical Projection

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $a$ be a set and $R$ an equivalence relation on $a$, and for $u\in a$ let $[u]_{R}$ be the equivalence class of $u$ under $R$. The two classes below are formed by class abstraction with the parameters $a$ and $R$; their formulas quantify over set variables only and are written with $[u]_{R}$ and the ordered pair as defined set symbols, so they are predicative as Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §comprehension requires.

¶ The quotient of $a$ by $R$ is the class

$$a/R=\{c:\exists u\,(u\in a\wedge c=[u]_{R})\}.$$

¶ The canonical projection of $a$ to $a/R$ is the class

$$\pi_{R}=\{p:\exists u\,(u\in a\wedge p=(u,[u]_{R}))\}.$$