hom-Sets

Let $\mathbb{C}$ be a fixed category.

Consider a morphism $f:A\to B$ in $\mathbb{C}$ then, for any object $X$ in $\mathbb{C}$, $\operatorname{hom}(X,1_A):\operatorname{hom}(X,A)\to\operatorname{hom}(X,A)$ is the identity map of $\operatorname{hom}(X,A)$ since for any $a\in A$, $$\operatorname{hom}(X,1_A)(a)=1_Aa=a.$$

Given $g:B\to C$ a morphism in $\mathbb{C}$, we have that $$\operatorname{hom}(X,g)\operatorname{hom}(X,f)(a) = \operatorname{hom}(X,g)(fa) = gfa = \operatorname{hom}(X,gf)(a)$$ and so $\operatorname{hom}(X,g)\operatorname{hom}(X,f) = \operatorname{hom}(X,gf)$. Similarly, for a morphism $c\in\operatorname{hom}(C,X)$ we have that, $$\operatorname{hom}(f,X)\operatorname{hom}(g,X)(c) = \operatorname{hom}(f,X)(cg) = cgf = \operatorname{hom}(gf,X)(c)$$ and so $\operatorname{hom}(f,X)\operatorname{hom}(g,X) = \operatorname{hom}(gf,X)$.

Consider morphisms $f:A\to B$ and $g:C\to D$ in $\mathbb{C}$, we can define the induced map $\operatorname{hom}(g,f):\operatorname{hom}(D,A)\to\operatorname{hom}(C,B)$ for $d\in\operatorname{hom}(D,A)$ by $$\operatorname{hom}(g,f)(d)=fdg$$ and we have that $$\operatorname{hom}(g,f)=\operatorname{hom}(C,f)\operatorname{hom}(g,A).$$

For a morphism $f$ in $\textbf{Sets}$ we have that $f$ is a monomorphism if and only if $\operatorname{hom}(X,f)$ is injective for every object $X$ in $\textbf{Sets}$. Suppose $f:A\to B$ is a monomorphism in $\textbf{Sets}$, then for any object $X$ in $\textbf{Sets}$ and any pair of parallel morphisms $a,a’\in\operatorname{hom}(X,A)$, we have that $$\operatorname{hom}(X,f)(a)=\operatorname{hom}(X,f)(a’)\implies fa=fa’ \implies a = a’$$ since $f$ is a monomorphism. So, $\operatorname{hom}(X,f)$ is injective for any object $X$ in $\textbf{Sets}$. Conversely, suppose $\operatorname{hom}(X,f)$ is injective for any object $X$, then for any pair of parallel morphisms $a,a’\in\operatorname{hom}(X,A)$, $$\operatorname{hom}(X,f)(a)=\operatorname{hom}(X,f)(a’)\implies a = a’$$ and so $$fa=fa’\implies a=a’$$ hence, f is a monomorphism in $\textbf{Sets}$. Similarly, $f$ is an epimorphism in $\textbf{Sets}$ if and only if for every object $X$ the map $\operatorname{hom}(f,X)$ is injective.