Suppose that the category $\mathbf C$ has powers and copowers of every object. Fixed a set $x$ one can define the functor $F:\mathbf C\to \mathbf C$, whose object function is $c\mapsto \coprod_x c$ (to be precise I should write $\coprod_{i\in x} (c)_i$); dually one can define the functor $G:\mathbf C\to \mathbf C$, whose object function is $c\mapsto \prod_x c$. This functions extend functorially on the arrows, and one can prove that there is a bijection $\mathbf C(F-,-)\cong\mathbf C(-,G-)$ natural in both variables; it follows that $F\dashv G$.
Fixed an object $c$ in $\mathbf C$ one can also define the functors $F':\mathbf {Set}\to \mathbf C$ extending $x\mapsto \coprod_x c$, and $G':\mathbf {Set}\to \mathbf C$ extending $x\mapsto \prod_x c$. One can prove that ${\mathbf C} (\coprod_x c,d) \cong \prod_x {\mathbf C}(c,d)$ naturally for every $d$ in $\mathbf C$ and set $x$. Also, fixed $x$ and letting $c$ vary in $\mathbf{C}$, using the functors defined in the first paragraph, the bijection above becomes natural in $c$ and $d$. These concepts all together confuse me a bit, can one say something general about this situation? I feel like there are some links that I am not seeing, but I have not really a precise question. Thanks for your patience