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\homework{Existentials}

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\begin{exercise}
Prove that the existential subtyping relation for a category~$\cat{Bnd}$ and functor $\cat{Bnd} \mto \cat{Prost}$ is always a preorder.
Be detailed in your proof.
\end{exercise}

\begin{exercise}
Prove that $\cat{Prost}$ has an $(\alg{E}, \alg{M})$ factorization structure, where $\alg{E}$ is the set of all epimorphisms and $\alg{M}$ is the set of all \emph{initial} mono-sources.
A $\cat{Prost}$-source $(\ob{C} \xmto{\mo{f}_i} \ob{C}_i)_{i : I}$ is initial when for all pairs $c_1, c_2 : C$, if $\forall i : I.\; f_i(c_1) \leq f_i(c_2)$ holds then $c_1 \leq c_2$ holds as well (note that the reverse implication always holds because each function must be relation-preserving).
You may assume that $\cat{Set}$ has a epi-mono factorization structure, that a morphism in $\cat{Prost}$ is an epimorphism iff its underlying function is an epimorphism, and that a source in $\cat{Prost}$ is a mono-source iff its underlying source is a mono-source.
\end{exercise}

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