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Let $c$ be an object of $C$ and $\mbox{const}_c: C\rightarrow C$ be the functor which sends every thing to $id_c$ and $\mbox{Id}_C:C\rightarrow C$ be the identity functor. Then $c$ is an intial object of $C$ iff there is a $\phi:\mbox{const}_c \Rightarrow \mbox{Id}_C $ s.t. component $\phi_c$ is an isomorphism. How to prove this? That such a natural transformation exists whenever $c$ is an initial object is hanksclear but I think I would like to see a proof for the other direction.


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up vote 3 down vote accepted

By definition $\phi$ is a family of morphisms $\phi_x : c \to x$ such that $\phi_c$ is an isomorphism and $f \circ \phi_x = \phi_y$ for every $f : x \to y$. In particular, for every morphism $f : c \to x$, we have $f \circ \phi_c = \phi_x$, hence $f = \phi_x \circ \phi_c^{-1}$. This shows that there is a unique morphism $c \to x$. Hence, $c$ is initial.

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