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Are there any special term for the following?

A function from the set of morphisms of a category to the set of morphisms of an other category preserving source and destination of every morphism.

I imply that the sets of morphisms of the two categories are the same.

Note that my functions are not functors, not even prefunctors.

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I imply that the sets of morphisms of the two categories are the same. –  porton Jul 13 '11 at 17:58
    
Please, add all relevant information to the body of the question, Porton.It is better if people reading do nothave to read all comments to find out what you implied. –  Mariano Suárez-Alvarez Jul 13 '11 at 18:02
    
So you are just dropping the requirement that a functor preserves the identity and respects compositions? –  Willie Wong Jul 13 '11 at 18:04
    
Willie Wong♦: It is just not a functor. –  porton Jul 13 '11 at 18:12

1 Answer 1

Morphism of graphs. ${}{}{}{}{}{}{}{}$

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A morphism of graphs may not preserve source or destination. –  porton Jul 13 '11 at 18:14
1  
@porto: a morphism of graphs which is the identity on obejcts... Not everything has a name. –  Mariano Suárez-Alvarez Jul 13 '11 at 20:19

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