fully faithful functor as conservative functor

Let \(\mathcal{C}, \mathcal{D}\) be categories \(\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}\) a fully faithful functor.

Then \(\mathcal{F}\) conservative functor.

Suppose \(\mathcal{F}: A \rightarrow B\) is an isomorphism in \(\mathcal{D}\). Then there exists an inverse morphism \(g\) By assumption, there exists \(\mathcal{F}^{-1}(f), \mathcal{F}^{-1}(g)\). Fconsrevatiurthermore, \(g \circ f = \mathrm{id}\) and thus

\begin{align*} \mathcal{F}^{-1}(g) \circ \mathcal{F}^{-1}(f) \vert \mathcal{F} \\ \mathcal{F}(\mathcal{F}^{-1}(g) \circ \mathcal{F}^{-1}(f)) =& \mathcal{F}(\mathcal{F}^{-1}(g)) \circ \mathcal{F}(\mathcal{F}^{-1}(f)) \\ =&g \circ f \\ =& \mathrm{id} \end{align*}

by uniqueness, we conclude, that

\begin{align*} \mathcal{F}^{-1}(g) \circ \mathcal{F}^{-1}(f) = \mathrm{id} \end{align*}