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I want these functors to have the following properties, they seem a bit arbitrary though - so I was looking for sufficient "standard" properties of functors which imply them (such as full, faithful etc.)

Let $F:C \to D$ be a functor with the following property. If $F(c)=F(c')$ then there exists a $f: c \to c'$ in $C$ such that $F(f)=\text{id}_{F(c)}$.

Let $G:C \to D$ be a functor with the following property. For all $f:c \to c'$ in $C$, if $G(f)=\text{id}_{F(c)}$ then $f= \text{id}_{c}$.

Thanks!

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Full implies the first property. Faithful does not quite imply the second, because different objects can be mapped to the same object. –  Arturo Magidin Jun 8 '12 at 17:06
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The second one is basically a weak version of faithful + injective on objects. A strong form of that might be "monomorphism in the category of categories". –  Zhen Lin Jun 9 '12 at 0:06
    
Thanks, this is very useful - in hindsight full should of been obvious for the first property. Unfortunately I can't use that $G$ is injective on objects, although I would like $G$ to be faithful if possible. –  Alexander Jun 9 '12 at 13:49
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up vote 2 down vote accepted

The first question has been answered in the comments.

Regarding the second question: the property for G that you describe is called "reflection of identities". You can find in the book "The joy of cats" section 13.36 that a sufficient condition for it is that G "creates isomorphisms".

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