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\begin{document}
\section{Yoneda lemma}
Let us introduce first a basic lemma in \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} that links the equivalence of two \htmladdnormallink{abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} to certain \htmladdnormallink{fully faithful functors}{http://planetphysics.us/encyclopedia/FullyFaithfulFunctor2.html}.
{\bf Abelian Category Equivalence Lemma.}
{\em Let $\mathcal{A}$ and $\mathcal{B}$ be any two Abelian categories, and also let $F: \mathcal{A} \to \mathcal{B}$ be an exact, fully faithful, essentially \htmladdnormallink{surjective}{http://planetphysics.us/encyclopedia/BCConjecture.html} \htmladdnormallink{functor}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. faithful, essentially surjective functor. Then $F$ is an equivalence of Abelian categories $\mathcal{A}$ and $\mathcal{B}$}.
The next step is to define the hom-functors. Let ${\bf Sets}$ be the \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of sets. The functors $F: \mathcal{C} \to {\bf Sets}$, for any category
$\mathcal{C}$, form a \htmladdnormallink{functor category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} ${\bf Funct}(\mathcal{C},{\bf Sets})$
(also written as $[\mathcal{C},{\bf Sets}]$. Then, any \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $X \in \mathcal{C}$ gives rise to the functor
$hom_C (X,−) : \mathcal{C} \to {\bf Sets}$. One has also that the assignment
$X \mapsto hom_C (X,−)$ extends to a natural contravariant functor
$F_y: \mathcal{C} \to {\bf Funct}(\mathcal{C},{\bf Sets})$.
One of the most commonly used results in category theory for establishing an equivalence of categories is provided by the following \htmladdnormallink{proposition}{http://planetphysics.us/encyclopedia/Predicate.html}.
{\bf Yoneda Lemma.}{\em The functor $F_y: \mathcal{C} \to {\bf Funct}(\mathcal{C},{\bf Sets})$ is a
\htmladdnormallink{fully faithful functor}{http://planetphysics.us/encyclopedia/FullyFaithfulFunctor2.html} because it induces \htmladdnormallink{isomorphisms}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} on the Hom sets.}
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