nLab
induction

In Set, the initial algebra for the functor $F:X\to 1+X$ is $⟨0,s⟩:1+\mathbb{N}\to \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers, $0$ is the smallest natural number, and $s$ is the successor operation. The principle of induction states that there is no proper subalgebra of $\mathbb{N}$; that is, the only subalgebra is $\mathbb{N}$ itself. This follows from the general property of initial objects that monomorphisms to them are isomorphisms.

More generally, the corresponding property of any initial algebra may be called induction. We then have induction over lists, trees, terms in a logic, and so on.