Every object admits an anodyne morphism to a saturated object

Can someone help me with the following problem

We call a morphism anodyne if it is both monic and epic.

We call an object B saturated if for every anodyne morphism $$A\xrightarrow{f} A'$$ and morphism $$A\xrightarrow{g}B$$,there is a morphism $$A'\xrightarrow{h} B$$ such that $$hf=g$$.

Now suppose that $$\mathcal{C}$$ is complete and well-powered, and that every object A of C admits a monomorphism $$A\rightarrowtail B$$ with B saturated. Show that every object admits an anodyne morphism to a saturated object [hint: consider the smallest strong subobject of B which contains A].

Let $$A$$ be an object in $$\mathcal{C}$$ and let $$B$$ be a saturated object with a morphism $$A\rightarrowtail B$$. Because $$\mathcal{C}$$ is well-powered, the strong subobject of $$B$$ that contain $$A$$ form a set $$\{A_i\rightarrowtail B\mid i \in I\}$$. This set induces a small diagram $$D$$. Let $$(L, \lambda_i:L\to A_i)$$ be the limit of this diagram. Because the intersection of strong subobjects is a strong subobject, we find that $$L$$ is a strong subobject of $$B$$. Because a strong subobject of a saturated object is saturated, we see that $$L$$ is saturated.

Because $$(A\rightarrowtail A_i)_i$$ induces a cone over $$D$$, there is a unique morphism $$A\xrightarrow{\rho}L$$ such $$A\rightarrow A_i$$ factors through $$\rho$$ for all $$i$$. This implies that $$\rho$$ is monic.

Can someone help me prove that $$\rho$$ is also epic?

• In the statement of the problem, shouldn't $g$ be $A\to B$? May 29 '19 at 12:29
• Yes sorry, it should.
– user302980
May 29 '19 at 12:33

If we have $$x,y:L\to K$$ with $$x\rho=y\rho$$, then $$x=y$$ since otherwise the equalizer of $$x$$ and $$y$$ would be a regular, thus strong, proper subobject of $$L$$ through which $$\rho$$ factored. But a strong subobject of $$L$$ is a strong subobject of $$B$$, and factoring $$\rho$$ implies factoring $$A\rightarrowtail B$$.