lemma

draft/draft.tex

@@ -2298,18 +2298,19 @@ Having lax versions of a symmetric relator, allows us to have simulation relatio
 \begin{lemma}
 	Given objects $A$, $X$, and $Y$ in a category $\BC$, then we have:
 	\begin{gather*}
-		\Hom(X,Y)\iso \Hom(\Hom(A,X)\to\Hom(A,Y))
+		\Hom(X,Y)\iso \Hom(\Hom(A,X),\Hom(A,Y))
 	\end{gather*}
 \end{lemma}
 \begin{proof}
-	It is entailed by the Yoneda lemma. \todo{Finish! You may need cartesian closedness!}
-\end{proof}
-\begin{cor}
-	Given objects $X$ and $Y$ in a category $\BC$, then we have:
+	It is entailed by the Yoneda lemma. The contravariant version of the Yoneda's lemma says that given a functor $G\c\BC^\op\to\Set$ the following correspondance holds:
 	\begin{gather*}
-		\Hom(X,Y)\iso \Hom(\Hom(1,X)\to\Hom(1,Y))
+		GX\iso\Hom(\Hom(\argument,X),G)
 	\end{gather*}
-\end{cor}
+	If we substitute $G$ with $\Hom(\argument,Y)$ then we have the following:
+	\begin{gather*}
+		\Hom(\argument,Y)\iso\Hom(\Hom(\argument,X),\Hom(\argument,Y))
+	\end{gather*}
+\end{proof}
 %
 \begin{prop}
 	Assuming the axiom of choice, given $(X_1 \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}X_2)$ and $(Y_1 \stackrel{q_1}{\leftarrow} S \stackrel{q_2}{\to}Y_2)$, there is a morphism $(g_1,g_2,w)\c(X_1 \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}X_2)\to(Y_1 \stackrel{q_1}{\leftarrow} S \stackrel{q_2}{\to}Y_2)$ in $\spa(\BC)$ iff there is a morphism $(g_1,g_2)\c(X_1 \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}X_2)\to(Y_1 \stackrel{q_1}{\leftarrow} S \stackrel{q_2}{\to}Y_2)$ in $\spa(\BC)$.