Categorical way of making monos commute

Question

Is there a categorical way (in terms of diagrams, limits, lifting properties etc) to formulate the requirement that for every pair of monos $f,g:C \to D$ there should be an endomorphism $h:D \to D$, s.t. $h\circ f = g$? I found a couple of specific monos (regular, strict,strong,orthogonal..) but nothing seems to fit for my problem..

Answer 1

This is not a condition on monomorphisms at all; as formulated it's a condition on $C$ and $D$. Here is the most closely related thing I know.

A topological space $X$ is sometimes said to be homogeneous if for every $x, y \in X$ there is a homeomorphism $f : X \to X$ such that $f(x) = y$. This is the special case of your situation in $\text{Top}$ where $C$ is a point and $h$ is required to be an isomorphism. Most spaces are not homogeneous but, for example, all connected manifolds without boundary are.

In general this condition seems hard to satisfy. I don't expect there's anything particularly nice to say about it.