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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.2011-07-13
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    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.2011-07-13
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    So you are just dropping the requirement that a functor preserves the identity and respects compositions?2011-07-13
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    Willie Wong♦: It is just not a functor.2011-07-13

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

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    A morphism of graphs may not preserve source or destination.2011-07-13
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    @porto: a morphism of graphs which is the identity on obejcts... *Not everything has a name*.2011-07-13
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    Porton-Suárez functors. No it does.2015-08-31