What kind of homomorphism is this? Suppose I have two spaces $A$, $B$, with relations $R^A$, $R^B$ on them. Let $f$ be a surjective homomorphism $f$ between them. That is, for all $a_1,a_2\in A$, $(a_1 R^A a_2 \Rightarrow f(a_1) R^B f(a_2))$, and $f:A\rightarrow B$ is onto.
I want the following additional requirement, and am asking if there is a name for spaces or maps that follow this requirement:
$(\forall b_1,b_2 \in B)[b_1 R^B b_2 \Rightarrow (\exists a_1, a_2\in A)[f(a_1)=b_1$, $f(a_2)=b_2$, and $a_1 R^A a_2]]$.
In words, if $b_1$ is $R^B$ less than $b_2$, then already there are $a_1, a_2$ such that $a_1$ is $R^A$ less than $a_2$, and $f(a_1)= b_1, f(a_2) = b_2$.
If $R^B$ is the induced relation — the minimal relation $R^B$ such that $a_1 R^A a_2 \Rightarrow f(a_1) R^B f(a_2)$, then I believe that requirement holds. However, I wonder if it holds in more general cases, and if there is a name for such a thing?
 A: Some people apparently call a homomorphism (not necessarily surjective) satisfying this condition a "strong homomorphism". See this question and my answer there. As explained in the linked question, the terminology "strong homomorphism" can also refer to a homomorphism which reflects relations: for all $\overline{a}\in A$, $R^A(\overline{a})\Leftrightarrow R^B(f(\overline{a}))$. So I think it's better to avoid this terminology.
The answer of Tristan Bice in the linked question suggests "weakly reflective" for your condition (and "strongly reflective" for the other meaning of "strong homomorphism"). This terminology seems entirely sensible to me, but I'm not sure  whether it exists in the literature.
Ok, now let's specialize to the surjective homomorphisms satisfying your condition. In the context of "Projective Fraïssé Theory", as developed by Trevor Irwin, Slawomir Solecki and other students and collaborators of Solecki's, these homomorphisms are called epimorphisms. See this paper for instance: Projective Fraïssé limits and the pseudo-arc by Irwin and Solecki. This name is also a bit unfortunate, because epimorphisms in the categorical sense need not satisfy this property. But at least it's a standard name which appears in lots of published papers.
Finally, I think the best name for a surjective homomorphism satisfying your condition would be a "quotient map" or possibly a "relational quotient map". This terminology appears, for example, in this paper: Dual Ramsey theorems for relational structures by Dragan Mašulović.
Why? Well, for any relational structure $A$ and any equivalence relation $E$ on $A$, we can form a quotient relational structure $A/E = \{[a]_E\mid a\in A\}$, by interpreting an $n$-ary relation symbol $R$ as follows: $R^{A/E}([a_1]_E,\dots,[a_n]_E\} \Leftrightarrow \text{there exist }b_1Ea_1,\dots,b_nEa_n\text{ such that }R^A(b_1,\dots,b_n)$. Note that there is a canonical quotient map $q\colon A\to A/E$, $q(a) = [a]_E$. And if $f\colon A\to B$ is any quotient map, then $f$ induces an equivalence relation $E_f$ on $A$ by $aE_fa'\Leftrightarrow f(a) = f(a')$, and there is a unique isomorphism $\varphi\colon A/E_f\to B$, given by $\varphi([a]_{E_f}) = f(a)$, such that $\varphi\circ q = f$. So, up to unique isomorphism, any quotient map is the canonical quotient map onto a quotient structure. [This is exactly like the situation with quotient maps of topological spaces.]
The reason I might suggest "relational quotient map" instead of just "quotient map" is because the construction above describes the quotient operation for the class of all relational structures (in a particular relational language), but not necessarily in any restricted class. For example, the quotient of a poset by an arbitrary equivalence relation may no longer be a poset (transitivity may fail). There's a natural quotient construction for posets, and the canonical map from a poset to its quotient deserves to be called a quotient map (of posets), but it doesn't always satisfy your condition. So when there's any chance for confusion, adding the word "relational" could indicate to your reader that the quotient is being carried out in the category of relational structures, not some subcategory.
