User:Austrodata/Definition:Restriction of Functor

Let $\mathbf A$ and $\mathbf B$ be subcategories.

Let $F: \mathbf A \to \mathbf B$ be a functor.

Let $\mathbf S$ be a subcategory of $\mathbf A$.

The restriction of $F$ to $\mathbf S$, denoted by $F|_{\mathbf S}$, is a functor $F|_{\mathbf S}: \mathbf S \to \mathbf B$ such that:

			Objects:			For each object $X$ of $\mathbf S$:		$\map {F\vert_{\mathbf S} } X = \map F X$
			morphism:			For each morphism $f: X \to Y$ of $\mathbf S$:		$\map {F\vert_{\mathbf S} } f = \map F f$
			Compatibility:			For each pair of composable morphisms $f, g$ in $\mathbf S$:		$\map {F\vert_{\mathbf S} } {g \circ f} = \map F {g \circ f} = \map F g \circ \map F f = \map {F\vert_{\mathbf S} } g \circ \map {F\vert_{\mathbf S} } f$