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](https://www.reddit.com/r/learnmath/comments/3bjenr/category_theory_isomorphismpreserving_functor/) before, but I don't see a satisfactory answer. 


Thanks for the coming help!



functors reflecting "isomorphism relations"?
Some authors name this "isomorphism reflecting", for example in [Noncommutative rings and their applications](https://bookstore.ams.org/conm-634) (p. 153) and [Models, Modules and Abelian Groups: In Memory of A. L. S. Corner](https://www.degruyter.com/viewbooktoc/product/37731) (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](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html) (p.8) and [Homology and homotopy in semi-abelian categories](https://arxiv.org/abs/math/0607100v1) (p.9), also [the nlab](https://ncatlab.org/nlab/show/conservative+functor).


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".


