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Given a category C, its morphism category D means a category that has

1) "morphisms of C" as its objects

2) "pairs (f,g) s.t. the diagram (square) commutes" as its morphisms.

The above definition is vague but I'm sure that most readers already know what 'Morphism category' (also called 'Arrow category') is. Here's my question.

If C = 'Category of vector spaces', does there exist the categorical (co)product of 'Morphism category'?

My guess is that the answer is negative and I've been searching for this on the internet, but couldn't figure it out. Would anyone help me with this (prove or give counterexample) or give me a reference? Since I'm a beginner in a category theory, I would be pleased if there is an elementary explanation. Thank you very much.

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    $\begingroup$ If $\mathcal{C}$ has an initial object and a terminal object, then the arrow category of $\mathcal{C}$ has exactly the same limits and colimits as $\mathcal{C}$, and they are calculated componentwise. $\endgroup$ – Zhen Lin Oct 7 '13 at 13:02
  • $\begingroup$ @Kyeong4 you're looking for a product in the morphism category or a product of morphism categories? $\endgroup$ – Giorgio Mossa Oct 7 '13 at 13:35
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Let $\mathcal C$ be a category with products $\def\Arr{\mathsf{Arr}}$$\Arr(\mathcal C)$ its arrow category. Given two objects, $f,g \in \Arr(\def\C{\mathcal C}\C)$, $f\colon C_f \to D_f$, $g\colon C_g \to D_g$, let $\pi_{i,C} \colon C_f \times C_g \to C_i$, $\pi_{i,D} \colon D_f \times D_g \to D_{i}$, $i \in \{f,g\}$ denote the product in $C$. Let $\Pi = f\times g \colon C_f \times C_g \to D_f \times D_g$ be the product morphism. Then $(\pi_{i,C}, \pi_{i,D}) \colon \Pi \to i$ is a morphism in $\Arr(\C)$, one can easily check, its the product.

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Let $\mathbf{2}$ be the category with two objects and one morphism between them.

Then the arrow category of any category $\mathbf{C}$ is isomorphic to the functor category $\text{Funct}(\mathbf{2}, \mathbf{C})$; consequently, limits and colimits can be computed from the general facts about limits of functors.

In particular, if $\mathbf{C}$ has all (co)limits of a given shape, then so does every functor category $\text{Funct}(\mathbf{D}, \mathbf{C})$, and (co)limits are computed pointwise: e.g.

$$ \left( \lim_{j \in J} F_j \right)(X) = \lim_{j \in J} \left( F_j(X) \right) $$

Thus, in the arrow category,

$$ \left(A \xrightarrow{f} B\right) \times \left(C \xrightarrow{g} D \right) \cong \left( A \times C \xrightarrow{f \times g} B \times D \right)$$

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Vector spaces have a product. So, given any two morphisms $f:A\longrightarrow C$ and $g:B\longrightarrow D$, you have a product $f\times g:A\times B\longrightarrow C\times D$. You should check to see if this satisfies your definition for product in the arrow category. You can do the same for coproduct.

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