What is the product and coproduct of Morphism category (Arrow category)? 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.

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.
