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.

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