(category theory) A unique morphism corresponding to each object of a category, which has its domain equal to its codomain, and which composed with any morphism (with which it is composable) gives that same morphism.

(category theory) The terminal object of a comma category from a functor to a fixed object; or, dually, the initial object of a comma category from a fixed object to a functor.

(category theory) A morphism which is both a constant morphism and a coconstant morphism.

(category theory) A morphism which yields the same morphism when post-composed with either one of a parallel pair of morphisms.

(category theory) A morphism from an object to the product of that object with itself, which morphism is induced by a pair of identity morphisms of the said object.

In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over .

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.