# isomorphic objects in a category

Let $C$ be a category. I have proved that for each $X\in Obj(C)$, $C(-,X): C\longrightarrow S$, where $S$ is the category of sets, is a contravariant functor. How to prove the following further results?

(1). Let $F:C\longrightarrow S$ be a contravariant functor. For each $X\in Obj(C)$, let $H_x$ denote teh set of all natural transformations $C(-,X)\longrightarrow F$. Prove that the map $H_x\longrightarrow F(X)$ defined by $\tau\mapsto \tau(Id_X)$ is bijective.

(2). $X$ and $Y$ are isomorphic objects in $C$ if and only if $C(-,X)\simeq C(-,Y)$.

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Hint: (1) Find an inverse map. There is just one way to find it, just write down the definition of a natural transformation here. (2) follows from (1). –  Martin Brandenburg Oct 26 '13 at 9:03