faithful functor in a sentence

1. A full and faithful functor is necessarily injective on objects up to isomorphism.
2. Some authors define an "'embedding "'to be a full and faithful functor.
3. Some authors define an "'embedding "'to be a full and faithful functor that is injective on objects ( strictly ).
4. In practice, however, the choice of faithful functor is often clear and in this case we simply speak of the " concrete category " C " ".
5. There is a fully faithful functor from the category of abelian groups to "'Rng "'sending an abelian group to the associated rng of square zero.
6. It's difficult to find faithful functor in a sentence.
7. Forgetful functors are almost always Concrete categories have forgetful functors to the category of sets indeed they may be " defined " as those categories that admit a faithful functor to that category.
8. The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of " p "-torsion groups.
9. I have also shown that if \ mathcal { E } is a class of idempotents containing all identity morphisms of \ mathcal { C }, then there is a full & faithful functor I : \ mathcal { C } \ to \ mathcal { C } [ \ mathcal { E } ].