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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

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

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A morphism of graphs may not preserve source or destination. – porton Jul 13 '11 at 18:14
@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
Porton-Suárez functors. No it does. – PyRulez Aug 31 '15 at 0:48

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