#homomorphism

Cartesian functions send {A}→{B} with exactly one tail a↦ per a∈{A} connecting to each head ↦b∈{B}.

In other words B has to be equal size or smaller than A.

This is true mapping rings to rings, groups to groups, sets to sets, vector spaces to vector spaces, ... it's just a property of arrows really.

When mathematicians want to talk about "one-to-many" (using the database lingo) or "multimaps" (some stupid word I heard on Wikipedia which absolutely nobody anywhere ever thought was a good term), though, they're not left outside.

If you've got a bundle of arrows ⇶ with tails from {a₀, a₁, a₂, a₃} ⇶ {b₁₄}, then that's a bundle of tails all heading to the same place. If you "grab them all by the head"

So when mathematicians want to talk about a multimap, they use a preimage ƒ⁻¹. Let's say the kernel for example--it's "everything that gets thrown in the trash"---so if multiple things get thrownin the trash, 

(linear subspace / quotient / ring morphism kernel)

So this is how they can associate a bunch of stuff, to one point. For example every point on a manifold gets a tangent space. Maybe this is a vector space for example--which is a lot bigger than just one point.

First, going from Korean to Japanese, you will notice that the function (translation) is bijective (word-for-word) because there exists a map between the two sets (sentences) where each element (word or word-part) in the first set (Korean) is paired with exactly one unique element of the second set (Japanese), each element of the second set is paired with exactly one unique element of the first set, and there are no repeats. Moreover, the mapping preserves the relations between elements in the sets and is, therefore, a homomorphism. In other words, the translation preserves the grammatical part of speech of each word or word-part (i.e., the direct object remains the direct object). Thus, the translation of this sentence from Korean to Japanese is a grammatical isomorphism. Obviously, the same can be said of the Japanese-to-Korean translation.

On the other hand, the translation from Japanese to English is not even a proper function because there exist no elements (word or word-parts) in the codomain (English) for two elements in the domain (Japanese): the topic particle は and the direct object particle を. Likewise, in the translation from English to Japanese, there exist no elements in the codomain for two elements in the domain: the definite article "the" and the possessive adjective "my." Additionally, there are two elements in the domain that map onto two elements each in the codomain: "went" to 行き and ました, and "with" to と and 一緒に. There is no function, let alone isomorphism, between English and Japanese. By extension, there is none between English and Korean, either.