Given a function $f:X\to Y$ in category $\mathcal{C}$, one can construct the image as a factorisation $f=(e:I\hookrightarrow Y)\circ(g:X\to I)$ that is universal (initial) among all such factorisations.
This does seem like a universal property. But I can't figure out how this can actually be constructed as an initial object in a comma category, because there are morphisms both from and to the object.