# Definition:Composition Functor on Categories of Subobjects

## Definition

Let $\mathbf C$ be a metacategory.

Let $C$ and $D$ be objects of $\mathbf C$.

Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ and $\mathbf{Sub}_{\mathbf C} \left({D}\right)$ be the associated categories of subobjects.

Let $g: C \to D$ be a monomorphism of $\mathbf C$.

Then $g$ defines a composition functor $g_* : \mathbf{Sub}_{\mathbf C} \left({C}\right) \to \mathbf{Sub}_{\mathbf C} \left({D}\right)$:

 Object functor: $\ds g_* f := g \circ f$ The composition $\circ$ is taken in $\mathbf C$ Morphism functor: $\ds g_* a := a$

That it is in fact a functor is shown on Composition Functor on Categories of Subobjects is Functor.

The effect of $g_*$ is captured in the following commutative diagram:

$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", <0em,-8em>*+{D} = "D", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.4)*!/_.6em/{f'}, "C";"D" **@{-} ?>*@{>} ?<>(.4)*!/^.6em/{g}, "X";"D" **\crv{<-5em,-4em>} ?>*@{>} ?*!/^1.6em/{g_* f = \\ g \circ f}, "X2";"D" **\crv{<5em,-4em>} ?>*@{>} ?*!/_1.6em/{g_* f' = \\ g \circ f'}, \end{xy}$