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In Sketches of an Elephant this kind of morphism is called a cover (section A1.3). In case you don't like that, here are some other possibilities (although I don't know that I actually like any of the …

In May's paper The additivity of traces in triangulated categories he has the following things to say about this property. (At least, I think it is an equivalent property -- his notation is different …

This is the Cauchy completion of $\mathcal{C}$ as an $\mathrm{Ab}$-enriched category.

I would call those objects "codiscrete" or "co-0-truncated", since "discrete" and "0-truncated" are used for the dual property, e.g. here and here. It's equivalent to saying that $B$ is equivalent to …

(Indeed, in this case we have "$C = M\circ E$" as monads in $\rm Prof$ via a distributive law, as shown by Cheng --- this long postdates the terminology, but the underlying intuition was probably there …

As Peter said in a comment, "objectwise" is quite common and would probably be understood even without quotation marks, although it never hurts to define a term the first time you use it. I expect th …

I think one name for this is a simulation.

Zhen is right that you can think of it as a lax monoidal pseudofunctor. In this paper, I called an equivalent structure a "monoidal fibration".

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-isomo …

Of course, as Finn pointed out, there is controversy over the choice for natural transformations, but my views on that are clear at the nlab page so I'll just write using that terminology. …

One name that I have seen used is semiadditive category.