I am trying to prove that $\{C(A,-)\mid A\in \text{ob}\mathcal{C}\}$ is a detecting family for the functor category $[\mathcal{C},\mathbf{Set}]$. For that, I need to check that taking a natural transformation $\alpha \colon F\longrightarrow G$, if $\beta \colon C(A,-) \longrightarrow G$ factors uniquely as $\beta= \alpha \circ \gamma$ for some $$\gamma \colon C(A,-)\longrightarrow F$$ natural transformation, then $\alpha$ is bijective.
By Yoneda's lemma, $\beta$ corresponds bijectively to an element of $GA$; more concretely, to $y=\beta_{A}(1_{A})$ and $\gamma$ corresponds bijectively to $x=\gamma_{A}(1_{A})$. Since $\alpha \circ \gamma=\beta$, we have that $\alpha_{A}(x)=y$.
However, I do not know how to conclude that $\alpha_A$ is an isomorphism.
Can anyone help me, please?
Thank you.
The fact that (for each $A$) each $\beta$, equivalently element of $GA$, gives rise to a unique $\gamma$, equivalently element of $FA$, means we can Skolemize to get a function $\delta_A:GA\to FA$ that witnesses this statement. This will be a pre-inverse to $\alpha_A$ by definition, i.e. $\alpha_A(\delta_A(y))=y$, which makes $\alpha_A$ a surjection. The uniqueness constraint essentially is the statement that $\alpha_A$ is an injection. So $\alpha_A$ is a bijection for each $A$, and a natural transformation that's an isomorphism for each component is a natural isomorphism.