$\operatorname{Hom}(-,A) \cong \operatorname{Hom}(-,A') \Rightarrow A \cong A'$, proof verification

Question

This is from pg93, ex 4.1.27.

Let $\mathcal{A}$ be a locally small category, and let $A,A' \in \mathcal{A}$ with $\operatorname{Hom}(-,A) \cong \operatorname{Hom}(-,A')$. Prove directly that $A \cong A'$.

My thoughts:

Let $\eta$ denote the natural isomoprhism between the two functors. We have $$H(A,A) \xrightarrow{ \eta_A} H(A,A')$$ So $\eta_A(id_A): A \rightarrow A'$ is a candidate. Now I want to construct inverse. $$ H(A',A') \xrightarrow{\eta_{A'}} H(A',A)$$ Then the map $\eta^{-1}_{A'} (id_{A'}):A' \rightarrow A$. So I'd like to show that the composition is identity. By the naturality condition of $\eta$, from $H_A(A) \rightarrow H_{A'}(A')$ (and similarly on other direction) $$ \eta_A(id_A) \circ \eta^{-1}_{A'}(id_{A'}) = \eta_{A'}(\eta_{A'}^{-1}(id_{A'}))= id_{A'} $$ We deduce these two maps are inverses. Hence $A' \cong A$.

Is this correct? Or is there a neat way to see this?

Answer 1

You proof is correct and I do not believe that there are many alternatives.

Answer 2

According to the Yoneda lemma an arrow $f:A\to A'$ exists such that for every object $B$ the map $\eta_B:\mathsf{Hom}(B,A)\to\mathsf{Hom}(B,A')$ is prescribed by $h\mapsto f\circ h$.

Likewise an arrow $g:A\to A'$ exists such that for every object $B$ the map $\eta_B^{-1}:\mathsf{Hom}(B,A')\to\mathsf{Hom}(B,A')$ is prescribed by $h\mapsto g\circ h$.

Then $f\circ g=\eta_A\circ\eta_A^{-1}(1_A)=1_A$ and $g\circ f=\eta_{A'}^{-1}\circ\eta_{A'}(1_{A'})=1_{A'}$ proving that $A$ and $A'$ are isomorphic.