Let $\mathcal{C}$ be a (locally small) category and let $X$ and $X'$ be objects in $\mathcal{C}$.
(Definition $1$) We say that $f\colon X\to X'$ is a monomorphism if for any object $Y$ in $\mathcal{C}$, the map $$\operatorname{Hom}(Y, X)\to \operatorname{Hom}(Y, X')$$ given by $g\mapsto f\circ g$ is a one-to-one function.
(Definition $1'$) We say that $f\colon X\to X'$ is a monomorphism if there exists $g\colon X'\to X$ such that $g\circ f=\operatorname{Id}_X$.
Of course both definitions generalize one-to-one functions in the category of sets, but Definition $1$ is the one which is used in the literature. However I want to know why my definition, namely Definition $1'$, is not as useful as the first one to be adopted as the standard way of defining monomorphisms. As a related question, is there any relation between the two versions? Does one definition imply the other one?
My own guess is Definition $1$ is chosen because of Yoneda's lemma.
Similarly one can give the following definitions for an epimorphism:
(Definition $1$) We say that $f\colon X\to X'$ is an epimorphism if for any object $Y$ in $\mathcal{C}$, the map $$\operatorname{Hom}(X',Y)\to \operatorname{Hom}(X, Y)$$ given by $g\mapsto g\circ f$ is a one-to-one function.
(Definition $1'$) We say that $f\colon X\to X'$ is an epimorphism if there exists $g\colon X'\to X$ such that $f\circ g=\operatorname{Id}_{X'}$.
(Definition $1''$) We say that $f\colon X\to X'$ is an epimorphism if for any object $Y$ in $\mathcal{C}$, the map $$\operatorname{Hom}(Y, X)\to \operatorname{Hom}(Y, X')$$ given by $g\mapsto f\circ g$ is an onto function.
My questions are similar to the case of monomorphism of categories. My guess is that a reason for choosing Definition $1$ as the "right" definition of an epimorphism is "duality."
Any answers or comments are highly appreciated.
Edit I thought it might be a good idea to include two possible ways of defining an isomorphism in order to compare them.
(Definition $1$) We say that $f\colon X\to X'$ is an isomorphism if there exists $g\colon X'\to X$ such that $f\circ g=\operatorname{Id}_{X'}$ and $g\circ f=\operatorname{Id}_X$.
(Definition $1'$) We say that $f\colon X\to X'$ is an isomorphism if it is both a monomorphism (in the sense of Definition $1$) and an epimorphism (as in Definition $1$).
Monomorphisms are defined the way they are because that is what works in many concrete categories of interest: $\mathbf{Set}$, $\mathbf{Grp}$, $\mathbf{Ring}$, etc. Even $\mathbf{Top}$ is an example if you expect monomorphisms to be injective continuous maps (as opposed to, say, topological embeddings). Moreover, with this definition, the Yoneda embedding sends monomorphisms to componentwise injective natural transformations, which is as much as what one can hope for.
By constrast, epimorphisms are defined the way they are because we want duality. It is true that epimorphisms in $\mathbf{Set}$ are surjections. It is also true in some concrete categories like $\mathbf{Ab}$ and $\mathbf{Top}$, but it is not true in other concrete categories: for instance, in the category of Hausdorff spaces and continuous maps, the epimorphisms are precisely the continuous maps with dense image.
I should also mention that a morphism is a split epimorphism if and only if the Yoneda embedding sends it to a componentwise surjective natural transformation. Indeed, the "only if" direction is clear, and if $f : A \to B$ is a morphism such that $$\mathrm{Hom}(T, f) : \mathrm{Hom}(T, A) \to \mathrm{Hom}(T, B)$$ is surjective for every $T$, then taking $T = B$, we find there is a morphism $s : B \to A$ such that $f \circ s = \mathrm{id}_B$, as claimed.
Addendum. There are categories in which morphisms that are simultaneously monomorphisms and epimorphisms are not necessarily isomorphisms. The easiest example is $\mathbf{Top}$: a continuous map that is both injective and surjective is not necessarily a homeomorphism.