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Strong monics are defined as: A monic $m$ is strong iff every commutative square $mu=ve$, in $E$ with $e$ epi, has a diagonal i.e. there is a morphism $t$ such that $u=te$ and $v=mt$ in $E$. (where $E$ represents the category)

The problem is:
Why are they taking this definition of strong monics, like why the commutative square and the diagonal, which makes it so strong (There is one other equivalent definition given in the book and also in nLab using orthoginality that I haven't gone through).

In SET strong monics are same as monics (which is true for any topos), so nothing can be deduced by looking there. In a non trivial poset (or a Heyting algebra, which is a quasitopos and not a topos) $P$, any morphism is monic but only the identity morphisms are strong monic. I am still unable to get much motivation of giving such a definition of strong monic.

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Orthogonality is quite useful. For instance, you can prove that (in any category) a morphism that is both strong monic and epic is an isomorphism. – Zhen Lin Jun 12 '14 at 12:31

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