Thanks to all who've responded.
I've been asked for "... the name given in an arbitrary category to an object A for which every mono B----->A is an isomorphism."
Peter Johnstone has suggested that such an object (whose poset of subobjects reduces to just the object itself) is simply "minimal". And I'm embarrassed at having lost so much of my former grasp of Galois theory:
... whether "the inclusion Q >-------> R of the ring of rational numbers into that of real ones is a bimorphism, in the category Rng of rings with units and units preserving ring homomorphisms."
Reflexively I think: monic, yes; epic, no, as permuting any two independent transcendentals should extend to a non-identity automorphism of R over Q.
No! There is no "non-identity automorphism of R over Q." Fortunately, as Mike Barr and George Janelidze have pointed out, each permutation of a transcendence basis of R over Q extends to an injection, over Q, of R into its algebraic closure C, which is good enough. Cheers, and thanks again, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]