factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
In the 2-topos Cat, the pair of classes of morphisms (essentially surjective and full functors, faithful functors) form a factorization system in a 2-category. This factorization system can also be restricted to the (2,1)-topos Grpd.
In fact, an analogous factorization system exists in any 2-exact 2-category and any (2,1)-exact (2,1)-category, including any Grothendieck 2-topos or (2,1)-topos; see here.
When restricted to Grpd, this is the special case of the n-connected/n-truncated factorization system in the (∞,1)-topos ∞Grpd for the case that $(n = 0)$ and restricted to 1-truncated objects.
For $f : X \to Y$ a functor between groupoids, its factorization is through a groupoid $im_2 f$ which is, up to equivalence, given as follows;
objects are those of $X$;
a morphism $[\phi] : x_1 \to x_2$ is an equivalence class of morphisms in $X$ where $[\phi] = [\phi']$ if $f(\phi) = f(\phi')$.
More on this is at infinity-image – Of Functors between groupoids.
(eso+full, faithful)-factorization system