If $C$ has coequalizers, then any morphism which is right orthogonal to epimorphisms must automatically be a monomorphism.

If $C$ has kernel pairs and coequalizers of kernel pairs, then any morphism which is right orthogonal to epimorphisms must automatically be a monomorphism.

Every strong monomorphism is extremal; the converse is true if $C$ has pushouts.

Examples

A nice example of strong monomorphisms in a category are the subspace inclusions in the category of diffeological spaces. In this setting, any subset $Y$ of a diffeological space $X$ is again a diffeological space. If smooth, the inclusion $\iota:Y \rightarrow X$ is always a monomorphism, but it is a strong monomorphism if and only if $Y$ has “enough” plots, that is if $\varphi: U\rightarrow Y$ is a plot if and only if the composite $\iota\varphi: U\rightarrow X$ is a plot.

Let $C$ be the category whose objects are the integers $\mathbf{Z}$, and whose morphisms are generated from arrows

$s_i, t_i : i \to i+1$

subject to the relations

$s\circ s = s\circ t = t\circ s = t\circ t.$

Then the only epimorphisms and monomorphisms in $C$ are the identities, thus every map is right orthogonal to all epimorphisms but only the identities are strong monomorphisms.

See at quasitoposthis lemma. Note that the result for quasitoposes immediately implies the result for toposes, since all monomorphisms $i: A \to B$ in a topos are regular ($i$ being the equalizer of the arrows $\chi_i, t \circ !: B \to \Omega$ in