Editing Cantor's theorem (section)

==Generalizations==
[[Lawvere's fixed-point theorem]] provides for a broad generalization of Cantor's theorem to any [[Category (mathematics)|category]] with [[Product (category theory)|finite products]] in the following way:<ref name="LawvereSchanuel2009">{{cite book|author1=F. William Lawvere|author2=Stephen H. Schanuel|title=Conceptual Mathematics: A First Introduction to Categories|year=2009|publisher=Cambridge University Press|isbn=978-0-521-89485-2|at=Session 29|url-access=registration|url=https://archive.org/details/conceptualmathem00lawv}}</ref> let <math>\mathcal{C}</math> be such a category, and let <math>1</math> be a terminal object in <math>\mathcal{C}</math>. Suppose that <math>Y</math> is an object in <math>\mathcal{C}</math> and that there exists an endomorphism <math>\alpha : Y \to Y</math> that does not have any fixed points; that is, there is no morphism <math>y:1 \to Y</math> that satisfies <math>\alpha \circ y = y</math>. Then there is no object <math>T</math> of <math>\mathcal{C}</math> such that a morphism <math>f: T \times T \to Y</math> can parameterize all morphisms <math>T \to Y</math>. In other words, for every object <math>T</math> and every morphism <math>f : T \times T \to Y</math>, an attempt to write maps <math>T \to Y</math> as maps of the form <math>f(-,x) : T \to Y</math> must leave out at least one map <math>T \to Y</math>.