TITLE: Morphism that is not a mapping
QUESTION [6 upvotes]: I have encountered a statement in Lang's Algebra (Revised Third Edition, page 53) concerning morphisms of objects that seems strange to me: "In practice, in this book we shall see that most of our morphisms are actually mappings, or closely related to mappings."
I have always been under the impression that the terms 'mapping' and 'morphism' are synonymous in the context of categories. Perhaps it is just the case that Lang defines the two in a way that they disagree, but I can't find such an instance. Are they in fact different? If so, what is an example of a morphism that is not a mapping?

REPLY [6 votes]: One of my favorite "counterexamples" to preconceived notions about categories is matrix algebra:

The objects are natural numbers
The arrows are matrices ($\hom(m,n)$ is the collection of $n \times m$ matrices)
Composition of arrows is the matrix product