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If it is given that morphism from $X$ to $Y$ is "Unique up to unique isomorphism". I think thus implying that $X \to Y$ is actually $X \to X$ and the morphism is identity morphism. Is this assumption correct?

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  • $\begingroup$ ‘Up to unique isomorphism’ means for any pair of isomorphic objects $(X,Y)$, there is the unique isomorphism $X\to Y$, not that there is only one isomorphism between $X$ and anything. $\endgroup$
    – arseniiv
    Dec 19, 2017 at 11:51

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If I understand your question correctly then: no.

Think of a category with exactly two objects $X,Y$ and - next to the identities - exactly one morphism $X\to Y$ and exactly one morphism $Y\to X$.

Then $X$ and $Y$ are isomorphic and the isomorphism $X\to Y$ and its inverse are unique, but we do not have $X=Y$.

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