**Background:** In homological algebra, a *quasi-isomorphism* of chain complexes is a chain map$\phi:(C,d) \to (C',d')$ so that the induced map on homology $\phi\_\ast:H\_\ast(C,d) \to H\_\ast(C',d')$ is an isomorphism.


This is clearly a special instance of the following much more general phenomenon: given a functor $\mathcal{F}:\mathcal{C} \to \mathcal{D}$, between categories $\mathcal{C}$ and $\mathcal{D}$, a morphism in $\mathcal{C}$ is taken to an isomorphism in $\mathcal{D}$ by $\mathcal{F}$.



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> What is the standard terminology for a morphism of $\mathcal{C}$ so that its image under $\mathcal{F}$ is a (mono, epi, iso) morphism?
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I couldn't find a term for this in MacLane's book, but I'm sorry if this is standard stuff.



Terminology generalizing "quasi-isomorphism"
I don't think this is completely standard, so if I were going to use it I would explain it first, but a natural possibility is "$\mathcal{F}$-isomorphism" (-monomorphism, -epimorphism). 


Quasi-isomorphisms are in fact sometimes called $H\_\*$-isomorphisms, and similarly maps inducing isomorphisms on homotopy groups are sometimes called $\pi\_\*$-isomorphisms (the main example of this usage that I can think of is for symmetric spectra, where $\pi\_\*$-isomorphisms are not the same as stable equivalences and thus need their own name).


