composition of conservative functors as conservative functor

Let \(\mathcal{C}, \mathcal{D}, \mathcal{E}\) be categories and \(\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}\) and \(\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}\) be conservative functors Then their composition \(\mathcal{F}_2 \circ \mathcal{F}_1\) is also a conservative functor

Let \(f \in \mathrm{Mor}(\mathcal{C})\) be a morphism such that \((\mathcal{F}_2 \circ \mathcal{F}_1)(f)\) is an isomorphism. Then since \(\mathcal{F}_2\) conservative, \(\mathcal{F}_1(f)\) is an iso.

Since \(\mathcal{F}_1\) is conservative, \(f\) is an iso.

Hence \(\mathcal{F}_2 \circ \mathcal{F}_1\) is conservative