full conservative functor and isomorphic objects

Let \(\mathcal{C}, \mathcal{D}\) be categories, \(\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}\) be a full, conservative functor. Suppose for \(X,X' \in \mathrm{Ob}(\mathcal{C})\) it holds, that

\begin{align*} \mathcal{F}(X) \cong \mathcal{F}(X') \end{align*}

are isomorphic

Then so are \(X,X'\)

by assumption, there exists an isomorphism

\begin{align*} f: X \rightarrow& X' \\ \end{align*}

By fullness of \(\mathcal{F}\), there exists an \(f' \in \mathrm{Mor}(\mathcal{C})\) such that

\begin{align*} \mathcal{F}(f') =& f \end{align*}

and by conservativeness, \(f\) is already an isomorphism, thus \(X \cong X'\)