Let us suppose that $C$ is a category where small limits are representable and $D$ is a small category.
Let $F:Hom(D,C)\rightarrow C$ be the functor defined by $F(f)=\varprojlim f$.
Is it true that $F$ has a right adjoint constant functor? Can someone please explain this to me? What about the unit and co-units here?
Also, could someone please suggest me some nice introductory books on this stuff? I am finding it very confusing.
On
As Andreas already answered, $F$ has no right adjoint in general. Let's see why this is so on a simple example.
Take $C=\mathrm{Set}$, the category of sets and take $D$ a category with two objects, call them $*$ and $**$, and only identity morphisms. Then given an element $f \in \mathrm{Hom}(D, C)$ its limit is simply the product $f(*) \times f(**)$. Now limits and colimits in $\mathrm{Hom}(D, C)$ are given pointwise. Take $f, g : D \to C$ to be the constant $1$ (terminal object in $\mathrm{Set}$) functors. Then $F(f + g) = (1 + 1) \times (1+1)$ which is a set of cardinality $4$. On the other hand $F(f) + F(g) = 1 \times 1 + 1 \times 1$ which is a set of cardinality $2$, thus $F(f + g) \not\cong F(f) + F(g)$.
Now if $F$ were to have a right adjoint it would itself be a left adjoint and left adjoints preserve colimits. $F$ does not preserve colimits, thus it has no right adjoint.
Note however that in a quite a few cases $F$ will preserve colimits and will have a right adjoint.
No, $F$ has no right adjoint in general, but it has a left adjoint, which sends each object $c$ of $C$ to the constant functor in $\text{Hom}(D,C)$ that sends every object of $D$ to $c$.
I'm probably old-fashioned, but I still think the best introductory book on category theory is Saunders Mac Lane's "Categories for the Working Mathematician".