Background: $\mathbf{Set}$ is a (cartesian) closed monoidal category, so we have the natural tensor-hom adjunction $\text{Hom}(X\times Y,Z)\cong\text{Hom}(X,\text{Hom}(Y,Z))$ for sets $X,Y,Z$.
Apparently, $\mathbf{Ab}$ is a "coclosed" category, i.e. there is an abelian group $\text{cohom}(A,B)$ such that $\text{hom}(\text{cohom}(A,B),C)\cong\text{hom}(A,B\times C)$ for abelian groups $A,B,C$.
I'm wondering whether $\mathbf{Set}$ is coclosed as well. Concretely, my questions are:
Is there a set, say $\text{Cohom}(X,Y)$, such that we have a natural isomorphism ("cohom-tensor-adjunction") $\text{Hom}(\text{Cohom}(X,Y),Z)\cong\text{Hom}(X,Y\times Z)$? How is it constructed?
Does $\text{Coend}(X):=\text{Cohom}(X,X)$ have any special properties? By duality, it should be a comonoid. Is the comultiplication different from the diagonal map?
Does the cohom-tensor-adjunction restrict to an isomorphism between "corepresentations" of $C$ and (right) $C$-comodules for a comonoid $C$? By a corepresentation of $C$, I mean a comonoid morphism $\text{Coend}(X)\to C$ for some set $X$.
There is a restricted naturality statement between $M$-linear isomorphisms and isomorphisms of representations of $M$ given by conjugation for a monoid $M$. Is there an analogue in the dualized setting? I.e. is there a naturality statement between some isomorphisms of corepresentations of $C$ and $C$-colinear isomorphisms for a comonoid $C$?
(If the construction of $\text{Cohom}(X,Y)$ is easy, I can probably figure out questions 2.-4. on my own.)