In section 5.2 of Topoi: the categorial analysis of logic by R. Goldblatt the epic-monic factorization of a Set function $f:A\to B$ is developed as follows:
We let $R_f:=\{(x,y)\mid f(x)=f(y)\}\subseteq A\times A$ together with the projections $p,q:R_f\to A$ be the pullback of $f:A\to B$ with itself. And now we consider $f_R:A\to A/R_f$ to be the coequaliser of the projections so that there is a unique $h:A/R_f\to B$ such that $f=h\circ f_R$.
The author then writes: This suggests that in a more general category we may attempt to factor an arrow by co-equalising its pullback along itsel. However, for technical reasons (the availability of results of the last section) it is simpler now to dualise the construction, i.e. to equalise the pushout of the arrow with itself.
So are these constructions the same in general? Is the co-equaliser of the pullback isomorphic to the equaliser of the pushout? By drawing the diagrams I managed to know that there is an arrow from the co-equaliser of the pullback to the equaliser of the pushout but nothing more. By now the book has already stablished that, in a topos, an arrow is iso iff it is both monic and epic. Can the arrow that I found be proven to be monic and epic?
Also, why does the author prefer this treatment? why dualise the construction? why not prove different results in the previous section? I feel that, by this point in the book pullbacks have been much more used to start using pushout now just to use the equaliser and not the co-equaliser. Is there a practical reason or is it only the author's preference?
Thanks!