Natural isomorphism of hom functors imply isomorphism of objects

Question

Let $\mathcal{C}$ be a category. Let $A$ and $B$ be objects of $\mathcal{C}$. If we have an isomorphism natural in $-$: $\mathcal{C}(-,A)\cong\mathcal{C}(-,B)$; does that imply $A\cong B$?

Answer 1

Yes. The Yoneda embedding $$y:\operatorname{Hom}_{\mathcal{C}}(A,B)\to\operatorname{Hom}_{\mathbf{Set}^{\mathcal{C}^{op}}}(\mathcal{C}(-,A),\mathcal{C}(-,B)),\quad y(X) = \mathcal{C}(-,X)$$ is fully faithful, and therefore reflects isomorphisms. Hence, given any natural isomorphism $\eta:\mathcal{C}(-,A)\cong \mathcal{C}(-,B)$, there exists an isomorphism $f:A\to B$ such that $y(f)=\eta$.