functors reflecting "isomorphism relations"? Consider a functor $F:C\to D$ between two categories $C$ and $D$. Suppose $F$ satisfies the following property: for any $a, b\in C$, $F(a)\cong F(b)\iff a\cong b$. 
Of course, $a\cong b\Rightarrow F(a)\cong F(b)$, so it is the other direction tricky. 
The question is then: is there a name for such functors? Have they been studied? 
I know that there are conservative functors, which reflects isomorphisms. However, the property of $F$ mentioned above only reflects "isomorphism relations." This question was asked here before, but I don't see a satisfactory answer. 
Thanks for the coming help!
 A: Some authors name this "isomorphism reflecting", for example in Noncommutative rings and their applications (p. 153) and Models, Modules and Abelian Groups: In Memory of A. L. S. Corner (p.480). But it is kind of dangerous to use this since some (maybe even more!) authors also mean "conservative" by "isomorphism reflecting", for example in Basic concepts of enriched category theory (p.8) and Homology and homotopy in semi-abelian categories (p.9), also the nlab.
I don't know if there is an English term for the isomorphism relation, in German it's just "Isomorphie" which could be translated to "isomorphy". Then the property should be called "isomorphy reflecting".
Of course, you could always just say "injective on isomorphism classes".
A: One paper about this is Elliott's Towards a theory of classification, where such functors are called classification functors, and there is a nice collection of examples.
(Though note that a category theorist would perhaps approach some of the material in that paper a bit differently, e.g. by considering inner automorphisms as 2-morphisms. And Elliott's claim that every category is concrete is wrong for size reasons.)
A: At several places in the book "Category Theory in context", E. Riehl calls this "$F$ creates isomorphisms" (in analogy to the more common term "$F$ creates (co)limits").
