undone Functions with left inverses are always injections. That is, given if there is a function such that for every , , then is injective. In this case, is called a retraction of Conversely, is called a section of Conversely, every injection with a non-empty domain has a left inverse . It can be defined by choosing an element in the domain of and setting to the unique element of the pre-image (if it is non-empty) or to (otherwise). The left inverse is not necessarily an inverse of because the composition in the other order, may differ from the identity on In other words, an injective function can be "reversed" by a left inverse, but is not necessarily invertible, which requires that the function is bijective. Injections may be made invertible In fact, to turn an injective function into a bijective (hence invertible) function, it suffices to replace its codomain by its actual range That is, let such that for all ; then is bijective. Indeed, can be factored as where is the inclusion function from into More generally, injective partial functions are called partial bijections. Other properties If and are both injective then is injective. If is injective, then is injective (but need not be). is injective if and only if, given any functions whenever then In other words, injective functions are precisely the monomorphisms in the category Set of sets. If is injective and is a subset of then Thus, can be recovered from its image If is injective and and are both subsets of then Every function can be decomposed as for a suitable injection and surjection This decomposition is unique up to isomorphism, and may be thought of as the inclusion function of the range of as a subset of the codomain of If is an injective function, then has at least as many elements as in the sense of cardinal numbers. In particular, if, in addition, there is an injection from to then and have the same cardinal number. (This is known as the Cantor–Bernstein–Schroeder theorem.) If both and are finite with the same number of elements, then is injective if and only if is surjective (in which case is bijective). An injective function which is a homomorphism between two algebraic structures is an embedding. Unlike surjectivity, which is a relation between the graph of a function and its codomain, injectivity is a property of