Let be a morphism $f : c \to d$ in a category $C$ is the image of the corresponding morphism in the opposite category $C^{op}$.
Find what are the coimages of these 2 situations
a) Given a Galois connection induced from a relation as in def 3.1, what is the coimage of that
closed elements of $P(X)$ are precisely those in the image $im(IE)$ of $IE$ ?
I want coimage from that conclusion, because they are still talking about the image.
b) Remember that in a quotient object $\ker$ (kernel pair) and $coeq$ (coequalizer) set up a Galois connection between $Quot(X)^{op}$ and $Rel(X)$ what is the coimage of the same
Galois connection between $Quot(X)^{op}$ and $Rel(X)$ ?
For the second situation, I think that coimage of galois connection between quotient and relation could be just a translation of opposite category from quotient to relation.
$Quot(X)$ and $Rel(X)^{op}$
In theory, it does not change the galois connection, but you should only move the opposite category between the 2 objects but I'm not sure and I don't understand the implication of this switch.
So if I understand correctly, you want to know what are the coimages of the closure operators associated to the two Galois connections you mention. But these are functions, so you want coimages in the category of sets : and these coincide with images, since every function can be factorized as a surjective one followed by an injective one, and these coincide with extremal (and even regular) epimorphisms an monomorphisms, respectively. So the coimages won't tell you anything more than the images already do.