Hi, I apologise if this is the wrong place for this question but i need to ask it somewhere.
The question is whether the right, (perhaps left?), adjoint to $Hom(-,A)$ exists in $\bf Set$ and how to deduse it.
I would very much like it to be the disjoint union $-\oplus A $ but im not quite sure how to deduce it or if its simply whishfull thinking.
Alternativly, is there an other adjoint (left or right) to the disjoint union.
I am writing an undergrad paper and I am aware of the coproduct. I'm asking since i hope the deduction of a dual adjoint will tell me a little of the general workings of category theory and help me understand how to make "computations" in concrete categories.
Also, if you have the time and will, is there a right adjoint to the covariant Hom functor?
Thanks in advance or sorry for wasting your time
Best Answer
To speak of right (or left) adjoints for a contravariant functor $F: C\to D$, one needs to decide whether to view it as a functor from $C^{op}$ to $D$ or as a functor from $C$ to $D^{op}$. What the one viewpoint calls a left adjoint, the other will call a right adjoint. One therefore often speaks instead of two contravariant functors being "adjoint on the right" (or on the left). In this language, $Hom(-,A)$ is adjoint on the right to itself, which means that morphisms from $X$ to $Hom(Y,A)$ are in natural bijective correspondence with morphisms from $Y$ to $Hom(X,A)$; here "natural" means (as usual for adjointness) with respect to both $X$ and $Y$.