Being a monomorphism described as a universal property Is it possible to describe the property "being a monomorphism" as a universal property (with appropriate category/ies and functor)?
 A: Well, it really depends on what you mean by "described as a universal property". Maybe you find this description acceptable:


*

*A morphism $f : A \to B$ is a monomorphism if and only if the diagram
$$\begin{array}{rcl}
A & \overset{\textrm{id}}{\rightarrow} & A \\
{\scriptstyle \textrm{id}} \downarrow & & \downarrow {\scriptstyle f} \\
A & \underset{f}{\rightarrow} & B
\end{array}$$
is a pullback square.
Indeed, suppose $g, h : X \to A$ are morphisms such that $f \circ g = f \circ h$; if the diagram is a pullback, then there is a unique $k : X \to A$ such that $g = h = k$, and so $f$ is a monomorphism; and if $f$ is a monomorphism then $g = h$ so there indeed a unique morphism completing the obvious diagram. 
This description is certainly useful. For example, it implies:


*

*Any functor that preserves pullbacks (or even just kernel pairs) also preserves monomorphisms. In particular, right adjoints preserve monomorphisms.

*Any functor that reflects pullbacks (or kernel pairs) and isomorphisms must also reflect monomorphisms. In particular, monadic functors reflect monomorphisms.
Or perhaps you would prefer something in terms of hom-sets:


*

*A morphism $f : A \to B$ in a (locally small) category is a monomorphism if and only if $f_* : \textrm{Hom}(X, A) \to \textrm{Hom}(X, B)$ is injective for all $X$.


If you think about it, this is just the definition of ‘monomorphism’. Amusingly this can be derived in a roundabout way by noting that $\textrm{Hom}(X, -)$ is a functor that preserves pullbacks, and the collection of all such functors jointly reflects pullbacks and isomorphisms.
