Hi Simon,
 

> I'd say this is (close to) a bug.
>

It is a bug, without qualification.
 

> So, we should think of being more precise, introduce "Set of partial maps 
> from 
> R to S", and use it as parents of partial maps that aren't maps. 
> Similarly, one should more consistently distinguish between the set of 
> all maps and the set of all morphisms. 
>

That seems to be the only reasonable solution.

I would like to add that in my opinion the first line of this example,

sage: f = GF(11).convert_map_from(GF(7))

should not be assumed to work at all, and it would be preferable to raise 
an error rather than returning something that is mathematically completely 
arbitrary.

Given the principle that no conditions are imposed on conversions (except 
that if a coercion exists, the conversion equals the coercion), it should 
in particular be perfectly fine if some of them do not satisfy the property 
of existing at all.

Peter

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