nLab
morphism

A morphism is what goes between objects in a category.

Given two objects in a category, say $x$ and $y$, there is a set $\mathrm{hom}(x,y)$, called a hom-set, whose elements are morphisms from $x$ to $y$. Given a morphism $f$ in this hom-set, we write $f:x\to y$ to indicate that it goes from $x$ to $y$.

The most familiar example is the category Set, where the objects are sets and the morphisms are functions. Here if $x$ and $y$ are sets, a morphism $f:x\to y$ is a function from $x$ to $y$.

More generally, a morphism is what goes between objects in any n-category.