Compatible Operations, Maps and Relations Pass to the Quotient

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 $a$; write $u\,R\,v$ for $(u,v)\in R$, $[u]$ for the equivalence class of $u\in a$, and $a/R$ for the quotient, a set by Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection §quotient-set.

¶ Let $\ast$ be a binary operation on $a$ such that, for all $u,u',v,v'\in a$, if $u\,R\,u'$ and $v\,R\,v'$ then $(u\ast v)\,R\,(u'\ast v')$. Then there is exactly one binary operation $\circledast$ on $a/R$ with $[u]\circledast[v]=[u\ast v]$ for all $u,v\in a$.

¶ Let $F:a\to a$ be a map such that, for all $u,u'\in a$, if $u\,R\,u'$ then $F(u)\,R\,F(u')$. Then there is exactly one map $\bar F:a/R\to a/R$ with $\bar F([u])=[F(u)]$ for all $u\in a$.

¶ Let $T$ be a relation on $a$ such that, for all $u,u',v,v'\in a$, if $u\,R\,u'$ and $v\,R\,v'$ then $(u,v)\in T$ if and only if $(u',v')\in T$. Then there is exactly one relation $\bar T$ on $a/R$ such that, for all $u,v\in a$, $([u],[v])\in\bar T$ if and only if $(u,v)\in T$.