Why every regular monomorphism is a monomorphism?

Question

I read in section 7.57 of Adameck's book (Abstract and Concrete Categories The Joy of Cats) that due to the uniqueness requirement in the definition of equalizer, every regular monomorphism must be a monomorphism. I see that in category theory, it is a usual practice to utilize the uniqueness property to prove equality theorems, but I could not use it here to prove that every regular monomorphism is left-cancellable.

Answer 1

Suppose $k:K\rightarrow X$ is a regular monomorphism, arising as the equalizer of two morphisms $f,g:X\rightarrow Y$.

We want to show that $k$ is leftcancelable, hence a monomorphism. To this end let $t,s: T \rightarrow K$ be two morphisms satisfying $kt=ks$. Call this composite $x:T\rightarrow X$. Now since $fx=fkt=gkt=gx$ the universal property of the equalizer $k$ yields a unique morphism $y:T\rightarrow K$ satisfying $x=ky$. But both $t$ and $s$ have this property, so the uniqueness in the universal property implies $t=y=s$.