The notion of a constant morphism in a category generalises the notion of constant function.
A constant morphism in a category is a morphism with the property that for any morphisms then .
Using the two-point set, it is simple to show that the constant morphisms in Set are precisely the constant functions.
As with , any morphism which factors through a terminal object is constant but although this is an “if and only if” in it need not be in a general category. It is, however, true in a regular category that any constant morphism factors through a subterminal object, namely its image.