In an abelian category, each subobject $A \stackrel{f}{\to} X$ individuate an equivalence relation $R(f) \to X^2$ which is given by the equalizer of $$X^2 \rightrightarrows X \to \text{Coker}(f). $$
In that case, this correspondence $\text{Sub}(X) \to \text{EqRel}(X)$ is even injective.
I am wondering if one can always find an injective map $\text{Sub}(X) \to \text{EqRel}(X)$. Of course, I do not believe that this is possible in general but in the case of Malcev categories, the following procedure might work.
Call $\delta : X \to X^2$ the diagonal, then $(f \times f) \vee \delta$ (the join of the two subobjects) is a natural candidate but I am not sure that it works.
Here we come with the question,
In a Malcev category is it possible to establish an injection $\text{Sub}(X) \to \text{EqRel}(X)$ in such a way that if $R(f) \to X^2$ splits, so does $A \stackrel{f}{\to} X$?
By split subobject I mean retract.
Of course, it is even better if Malcev is an unneeded hypothesis.