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What is a morphism?

What is a morphism? Here are some definitions.

Noun
  1. (mathematics) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.
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Examples
A groupoid is a category in which every morphism is an isomorphism.
The concept of coding is based on the new notion of strong morphism between trace monoids.
A trace coding is an injective morphism between two trace monoids.
The fact that each morphism has an inverse corresponds to the fact that identity is a symmetric relation.
They also genotyped each participant using single nucleoptide poly morphism analysis of MC1R alleles.
As we shall see below, this is not the case for algebraic fibring, unless a stronger notion of morphism between logics is adopted.

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