The codomain of a function $f : A \to B$ is the set B — the collection of possible outputs. Not every element of the codomain need actually appear as an output. Those that do form the image (or range) of f. A function is surjective when its image equals its entire codomain — every possible output is actually hit.

The codomain is part of the function’s specification, not a consequence of its rule. The function $f(x) = x^2$ viewed as $f : \mathbb{R} \to \mathbb{R}$ has codomain $\mathbb{R}$ and is not surjective (negative numbers are never outputs). The same rule viewed as $g : \mathbb{R} \to [0, \infty)$ has a smaller codomain and is surjective. These are different functions because they have different codomains, even though they assign the same output to every input.

The codomain of a binary relation $R \subseteq A \times B$ is the set B — the set from which the second components of the ordered pairs are drawn.

In a category, the codomain of a morphism $f : A \to B$ is the target object B. Composition requires the codomain of the first morphism to match the domain of the second: $g \circ f$ is defined only when the codomain of $f$ equals the domain of $g$. This matching condition is the basic constraint on which morphisms can be chained together.