Let and be functions in the Category , such that . We seek to prove that is injective under these conditions.

Injective Monomorphism

Assume that is Injective. Then it’ll follow that

Thus, is a monomorphism.

Monomorphism Injective

Let be a Monomorphism. Then for any two functions , it’ll follow that , for any functions .

The proof is “constructive”: Let and some . Then choose such that

that is, they are the same function, but they (may) differ for some .

Then we have per hypothesis that

In particular, we have that it follows for :

Since this can be done for all points , then is injective¹.

And, in fact, this proves that, given a monomorphism, we cannot construct as above such that they differ in a single point: the monomorphism condition ensures that they are necessarily equal to all points ─ provided that their composition with is equal. ↩