Whether the morphism has special name

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If morphism h is an extension of morphism f,we call morphism f is a restricion of morphism h; If morphism h is a lifting of morphism f,what do we call morphism f of morphism h ? Or in other words, if i is an inclusion morphism ,then what do we call the morphism f*i, as i*f is called restriction of f ?
 

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What you call it entirely depends on how one has lifted; there are normally going to be (infinitely) many different lifts with no reason to suppose that anyone is _the_ lift.

An example where one does have some special properties would be the induced morphism in representation theory from the induction functor. (One might well need some functor lying around to show that something exists and is unique and has the right universal properties.)