Power set functors preserve monicness This link discusses power set functors. 

Proposition 5.7 If $f$ is a epimorphism then so is $\exists_f$. 
Proposition 5.8 If $f$ is an monomorphism then so is $\forall_f$.

A little confused...doesn't proposition 5.8 apply to $\exists_f$ also? Certainly if $f$ is injective then $f^* \circ \exists_f = \text{Id}$, so $\exists_f$ must be injective.
Am I missing that they're talking about a richer category than just $\mathsf{Set}$? Maybe because of additional structure monic means more than just being injective here?
 A: Note the language at the top directory:

In these notes the general instructions are to prove or disprove and salvage if possible each of the Propositions and Theorems. Use the definitions as given.

I have no direct experience thinking of power set functions in this context, so I'm not sure what 'results' are actually true.
Digging in more specifically, the language 

Proposition 5.2$\;$ $f^*: \mathcal{P}(B) \to \mathcal P(A)$ is a functor; that is, if $B'' \subseteq B'$, then $f^*(B'') \subseteq f^*(B')$.

suggests that here a functor should preserve containment. So, I think what we have here are "categorified posets" (see here); that is, given a set $S$, its powerset $\mathcal{P}(S)$ is the category whose objects are subsets of $S$. For $s', s'' \in \mathcal{P}(S)$ we have a morphism $s' \to s''$, if and only if $s' \subseteq s''$. 
Viewed in this context, it is indeed asking you to show that $f^*$ preserves morphisms; that is, $B' \to B''$ implies $f^*(B') \to f^*(B'')$.
I'm not absolutely sure of this, but it seems to be the most sensible way to speak of functions between power sets as functors.
