What is the name for a function whose codomain and domain are equal?

Solution 1:

Whilst an endomorphism is a morphism or homomorphism from a mathematical object to itself, the technical term for a function that has a domain equal to it's co-domain is called an endofunction.

NB: A homomorphic endofunction is an endomorphism.

Edit: From Wikipedia:

Let $S$ be an arbitrary set. Among endofunctions on $S$ one finds permutations of $S$ and constant functions associating to each $x \in S$ a given $c \in S$.

Every permutation of $S$ has the codomain equal to its domain and is bijective and invertible. A constant function on $S$, if $S$ has more than $1$ element, has a codomain that is a proper subset of its domain, is not bijective (and non invertible). The function associating to each natural integer $n$ the floor of $n/2$ has its co-domain equal to its domain and is not invertible.

Finite endofunctions are equivalent to directed pseudoforests. For sets of size $n$ there are $n^n$ endofunctions on the set.

Particular bijective endofunctions are the involutions, i.e. the functions coinciding with their inverses.

Solution 2:

Endomorphisms: https://en.wikipedia.org/wiki/Endomorphism ${}{}{}{}{}{}{}$

Solution 3:

Another term that is sometimes used, especially in the context of topological spaces and related objects, is "self-map".