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 $x\mapsto b$.

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  • $\begingroup$ How to prove the converse? Just give me a hint. $\endgroup$ – user170039 Dec 11 '19 at 10:26
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    $\begingroup$ Consider $\zeta_A^{-1}(1_A)\in U(A)$. $\endgroup$ – Fabio Lucchini Dec 11 '19 at 10:36
  • $\begingroup$ Perfect! Thank you very much. $\endgroup$ – user170039 Dec 11 '19 at 14:23

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