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As in Explicit construction of a initial object in a topos I'm looking an elementary proof of the fact that, in a topos, epimorphisms are stable under pullback or, equivalently, that images are stable under pullback. Stardard proofs uses the fact that the pullback functor has right adjoint, hence it preserve colimit.

Proofs can assume the existence of initial object and the image of a morphisms. In particular I'm looking for a proof wich make uses of the internal logic as here: http://ncatlab.org/nlab/show/Trimble+on+ETCS+III

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  • $\begingroup$ Please state the definition of topos you are using, if you are not assuming that pullbacks have right adjoints. $\endgroup$ – Zhen Lin Sep 27 '13 at 21:26
  • $\begingroup$ I use the elementary definition of topos, namely a cartesian closed category with power objects. $\endgroup$ – Fabio Lucchini Sep 27 '13 at 21:28
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    $\begingroup$ You can find an "internal logic" proof in [Introduction to higher order categorical logic], Lemma 6.5. $\endgroup$ – Zhen Lin Sep 27 '13 at 21:35

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