What is the proper term for a function where domain and codomain coincide? > What is the proper term for a function where domain and codomain coincide? E.g. in programming languages a function f : Int => Int or f : Double => Double. Thanks.

Suppose we have a function $f:S\to S$, that is $f$ is a function from $S$ with image (range) in $S$. Such a function is called an ~~endomorphism~~ endofunction (endo comes from the greek or latin word for "within"). Sometimes people use the term codomain to mean the image (range) of a function. If $S$ is finite and the image of $f$ is the whole set $S$, then $f$ is a bijection. If $S$ is infinite, the situation is somewhat more complicated.

Edit: Thanks for the correction.