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In a category with binary products in a category, a map with the left lifting property against monomorphisms is an epimorphism. If $ue=ve$, lift the square with right side $\Delta$ and bottom side $⟨u,v⟩$. The two projections of its filler give $u=v$. This argument does not require equalizers.

If $e:X\to Y$ is a monomorphism and has the left lifting property against monomorphisms, apply its property to the square with both vertical arrows $e$ and both horizontal arrows identities. The filler is a two-sided inverse, so $e$ is an isomorphism.

Here use the left lifting property against monomorphisms as the definition of “strong”; epimorphicity will be established separately. In a lifting square write $g:C\to D$, $m:A\to B$, $k:C\to A$ and $l:D\to B$, with $mk=lg$ and $m$ a monomorphism. Any two lifts agree because $mt=m{t}^{\prime }=l$.

First let $g$ be the coequalizer of $r,s:E\rightrightarrows C$. Every coequalizer is an epimorphism: if $ug=vg$, the uniqueness clause for the coequalizer applied to this common composite gives $u=v$. Moreover,so $kr=ks$ by monomorphism cancellation. The coequalizer therefore supplies $t:D\to A$ with $tg=k$. Then $mtg=mk=lg$, and epimorphism cancellation gives $mt=l$. This proves that regular epimorphisms are strong epimorphisms.

If $g:C\to D$ is also a monomorphism, take $m=g$, $k=1_C$ and $l=1_D$. Its lift satisfies $tg=1_C$ and $gt=1_D$. Hence monic lifting-only strong morphisms are invertible.

Next suppose $gf:A\to C$ has the left lifting property against monomorphisms. Given a lifting square for $g:B\to C$, with $mk=lg$, precompose its top arrow with $f$. A lift for $gf$ gives $t:C\to X$ with $mt=l$ and $tgf=kf$. Crucially, one does not cancel $f$: instead $mtg=lg=mk$, and the monomorphism $m$ gives $tg=k$. Thus the right factor of a strong composite is strong, proving right-factor cancellation for lifting-only strong morphisms.

Finally, in $u=iv$ with $u$ strong and $i$ monic, the preceding result makes $i$ strong. The monic-strong argument then makes $i$ an isomorphism. None of these arguments assumed that a lifting-only strong morphism was already epic.

Let $g:C\to D$ have the left lifting property against monomorphisms, and suppose $\alpha ,\beta :D\to B$ satisfy $\alpha g=\beta g$. Use the categorical diagonal ${\Delta }_B:B\to B\times B$, which is a monomorphism by part (a). The square with top arrow $\alpha g$, bottom arrow $⟨\alpha ,\beta ⟩$, left arrow $g$ and right arrow ${\Delta }_B$ commutes, since both product components are $\alpha g$.

Its lift $t:D\to B$ satisfies ${\Delta }_{B}t=⟨\alpha ,\beta ⟩$. Applying the two product in a category projections gives $t=\alpha$ and $t=\beta$. Thus $g$ is an epimorphism. This binary-product criterion for lifting-only strong epimorphisms requires binary products, rather than any assumption about equalizers.

If $gf$ has the left lifting property against monomorphisms, so does $g$. Precompose a lifting square for $g$ with $f$ and lift the composite. To check the remaining triangle, cancel the monic side of the square, rather than cancelling $f$. In particular, a monic right factor of a lifting-only strong morphism is an isomorphism.