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Let $C$ be a category and $Mono\left( C\right)$ the category which has: $Ob \left(mono\left(C\right)\right)=\left\{u: ux_1=ux_2 \implies x_1 =x_2\right\}$

$mono\left(C\right)\left(u,v\right)=\left\{(a,b):va \square bu \right\}$ where the square notation says the obvious square is pull-back. Is $Mono\left( C\right)$ a fullsubcategory of $C^{\rightarrow}$ (the arrow category) i.e. the pullback property of the morphisms between two monomorphisms is necessary?

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The category you call $\mathbf{Mono}(\mathbf{C})$ is definitely not (usually) a full subcategory of $\mathbf{C}^\to$; an easy counterexample is to take any such pullback square under consideration, and replace its corner with a proper subobject.

For explicitness, suppose $f:A \to B$ is monic. Then

$$\begin{matrix} A &\to& B \\ \downarrow & & \downarrow \\ B &\to& B \end{matrix} $$

is a morphism in $\mathbf{C}^\to$ from $f$ to $1_B$.

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Thank you. Do you know if the full subcategory of $C^{\rightarrow}$ with monos as objects is of any application? I think the defenition is as such just because it's enable us to express the subobject classifier as the terminal of $mono\left(C\right)$. –  Hooman Apr 1 '13 at 3:17
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@Hooman, such categories show up often in the context of $K$-theory. –  Mariano Suárez-Alvarez Apr 1 '13 at 4:06

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