restriction of a functor

Let \(\mathcal{C}, \mathcal{D}\) be categories, \(\mathcal{E} \subseteq \mathcal{C}\) a subcategory and \(\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}\) a functor. then the restriction \(\mathcal{F}_{\mathcal{E}}\) is defined as functor

\begin{align*} \mathcal{F}_{\mathcal{E}}: \mathcal{E} \rightarrow& \mathcal{D} \\ A \mapsto& \mathcal{F}(A) \\ (f: A \rightarrow B) \mapsto& \mathcal{F}(f): \mathcal{F}(A) \rightarrow \mathcal{F}(B) \\ \end{align*}

properties of functors (identity, composition) are preserved, hence welldefined