Representable functors preserve limits. Theorem : Representable functors preserve limits.
I'm struggling to see why this is true. It's not obvious to me where I should be actually using the fact that functor in question is representable.
Any help would be much appreciated.
 A: Firstly we have to prove that $\hom(a,-)$ preserves limits.
Let $A$ be a category, $a\in A$ be its object and $\hom_A(a,-)\colon A\to\mathbf{Set}$ be a $\hom$-functor, corresponding to this object, let $B$ is also a category, $T\colon B\to A$ be a functor. Let's assume that $a_0$ is a limit of $T$, thus, we can consider a limiting cone $\varphi\colon\Delta_{a_0}\to T$. We want to prove that $\hom_A(a,a_0)$ is a limit of the functor $\hom_A(a,T(-))$ and that the induced cone $\varphi_*\colon\Delta_{\hom_A(a,a_0)}\to \hom_A(a,T(-))$ is also limiting. Let $X$ be an arbitrary set and $\psi\colon\Delta_X\to \hom_A(a,T(-))$ be an arbitrary natural transformation. Now we want to find a mapping $f\colon X\to \hom_A(a,a_0)$, such that $\varphi_*\circ\Delta_f=\psi$. For any element $x\in X$, let's construct a natural transformation $\psi_x\colon\Delta_a\to T$, such that $\psi_x(b)=(\psi(b))(x)$(check that this is a natural transformation). This construction will give you an arrow $f_x\colon a\to a_0$, such that $\varphi\circ\Delta_{f_x}=\psi_x$. Define the mapping $f$ by putting $f(x)=f_x$. Check that such mapping is unique which satisfies the property $\varphi_*\circ\Delta_f=\psi$.
Secondly, if $A$ and $B$ are categories, $T,S\colon A\to B$ are naturally isomorphic functors, then $T$ is continious(=preserves all limits) iff $S$ is continious(it is an easy exercise).
Now, if you have a representable functor, then it is isomorphic to some hom-functor, and therefore it is also continious - QED.
