# Free objects and representable functor

For the sake of completeness let me first begin with some definitions. I will be assuming the definitions of a coslice category.

Definition 1. Let $$(\mathbf{A},\mathscr{U})$$ be a concrete category over $$\mathbf{X}$$ and $$X$$ is an $$\mathbf{X}$$-object. A $$\mathbf{X}$$-morphism $$u:X\to\mathscr{U}(A)$$ is said to be universal iff $$(X,u,A)$$ is an $$(X\downarrow\mathbf{X})$$-initial object (where $$(X\downarrow\mathbf{X})$$ denotes the coslice category of $$\mathbf{X}$$ with respect to $$X$$.

Definition 2. Let $$(\mathbf{A},\mathscr{U})$$ be a concrete category over $$\mathbf{X}$$ and $$X$$ is an $$\mathbf{X}$$-object. An $$\mathbf{A}$$-object $$A$$ is said to be free over $$X$$ if there exists a universal $$\mathbf{X}$$-morphism $$u:X\to\mathscr{U}(A)$$.

The problem with which I am stuck is the following,

Let $$(\mathbf{A},\mathscr{U})$$ be concrete category over $$\mathbf{Set}$$. Let $$\{x\}$$ be a singleton set and $$A$$ be an $$\mathbf{A}$$-object. Then $$A$$ is free over $$\{x\}$$ iff $$\mathscr{U}$$ is naturally isomorphic to $$\operatorname{Hom}_{\mathbf{A}}(A,-)$$ (where $$\operatorname{Hom}_{\mathbf{A}}(A,-)$$ is the covariant $$\operatorname{Hom}$$-functor).

I have not found any way of constructing the required natural isomorphism yet. Can anyone help me?

Let $$u:\{x\}\to U(A)$$ be an $$U$$-universal morphism. For every object $$B$$ in $$\mathbf A$$ we have to define a bijection $$\zeta_B:U(B)\to\hom_{\mathbf A}(A,B)$$. For every $$b\in U(B)$$, $$\zeta_B(b)$$ is the only morphism $$A\to B$$ such that $$U(\zeta_B(b))\circ u$$ is the function $$\{x\}\to U(B)$$ such that $$x\mapsto b$$. • Consider $\zeta_A^{-1}(1_A)\in U(A)$. – Fabio Lucchini Dec 11 '19 at 10:36