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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?

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

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