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