So we have a category $C$ and objects $X$ and $Y$. Given that Mor$_C(-,X)$ and Mor$_C(-,Y)$ are isomorphic functors, then prove that $X\cong Y$ (as objects in $C$).
So by the condition I know that there exists a natural transformation $\eta$ such that for each object $A\in C$ we obtain the map $\eta_A$ such that $\eta_A:$Mor$_C(A,X)\rightarrow$ Mor$_C(A,Y)$. Further, we have that each $\eta_A$ is an isomorphism. My doubts comes on what $\eta$ does to Mor$_C(A,X)$, does it take an arrow from $A$ to $X$ and it makes it now point to $Y$? My intuition tells me that if we let $A=X$, then we have two arrows (call them $f$ and $g$) in Mor$_C(X,X)$ such that $fg=1_X$, and somehow I feel that $\eta_X$ should take $f$ and $g$ to arrows that do the same thing.
Could someone please tell me whether my intuition is correct and perhaps show me how to do this with rigor?