I'm trying to get my head around adjoints to the forgetful functor $U:\bf{C^M}\to \bf {C}$ where $M$ is a monoid interpreted as a category.
My current line of thinking is that the left adjoint $L:\bf C\to C^M$ is given by $L(A)=M\times A$ where $M$ acts on $M\times A$ by $m(M\times A)=(mM)\times A$, the unit $u:A\to M\times A$ is given by the inclusion $u(A)=(1,A)$ and the counit is $\varepsilon:M\times A\to A$ is given by $\varepsilon((m,A))=mA$
It seems to me that this works out quite well though I havent written the down the bijection. One problem that comes to mind is that $M\times A$ might not be an object in $C$, in which case it's not an object in the functor category, and I'm not quite sure how to address this.
Regarding the right adjoint, I'm thinking $R(A)=Hom(M,A)$ but I have almost no idea of how to proceed here. I know that we can interpet the $Hom(M,A)$ sets as objects in many categories, but it does not seem like a general phenomenon.
Anyways, any thought or comments would be helpful, thanks in advance
Edit1:
Ok, i think i got it for $\bf{Set}$. We let the unit $u_A:A\to Hom(M,A)$ be the map $u_A(a)=f_a:M\to A$ given by $f(M)=a$ and the counit $\varepsilon_B:Hom(M,B)\to B$ be given by $\varepsilon(f)=f(1)$. Chasing the diagram seems to work out fine and all so im guessing its alright, ganna write down the bijection in a min. I'm still a bit unsure as to when $Hom_{\bf C}(M,-) $ can be considered an object in $\bf C$ it works out fine in most, (all?), algebraic categories but beyonde this i have no idea.
Anyways, thanks for the replies, they were very helpful, though i have no idea as to why the forgetful functor should be both a tensor functor and a hom functor, some reference would be nice