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In the reaction

$$\ce{3CuO (s) + 2NH3 (aq) <=> N2(g) + 3H2O(l) + 3Cu(s)}$$

the oxidation number of $\ce{Cu}$ goes from $+2$ in the reactant side to $0$ in the products, and so it is reduced.
The oxidation number of $\ce{O}$ is $-2$ and stays the same on both sides.

My textbook refers to $\ce{CuO}$ as the oxidising agent. Is this because the $\ce{Cu}$ in $\ce{CuO}$ is oxidised while $\ce{O}$ is neither oxidised or reduced, so overall it is an oxidising agent?

What happens for a compound $\ce{AB}$ where $\ce{A}$ gets oxidised and $\ce{B}$ gets reduced? Do I call the compound $\ce{AB}$ a reducing agent if the net change in oxidation number (between the two elements $\ce{A}$ and $\ce{B}$) is negative?

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My textbook refers to $\ce{CuO}$ as the oxidising agent. Is this because the $\ce{Cu}$ in $\ce{CuO}$ is oxidised while $\ce{O}$ is neither oxidised or reduced, so overall it is an oxidising agent?

No. $\ce{CuO}$ is the oxidizing agent (the thing that causes something else to be oxidized) because the $\ce{Cu}$ is reduced (gains electrons, going from oxidation state $+2$ to $0$) in the course of oxidizing the nitrogen of $\ce{NH3}$, which loses electrons and goes from oxidation state $-3$ to $0$.

Conversely, $\ce{NH3}$ is the reducing agent here, because it causes something else to be reduced (here, the $\ce{Cu}$) and loses electrons in the process.

What happens for a compound $\ce{AB}$ where $\ce A$ gets oxidised and $\ce B$ gets reduced?

If only the species $\ce{AB}$ is involved, or in the case where two of the same species $\ce A$ react to form both oxidized and reduced products:

$$ \ce{2A -> Ox + Red} $$

the reaction is called disproportionation.

Do I call the compound $\ce{AB}$ a reducing agent if the net change in oxidation number (between the two elements $\ce A$ and $\ce B$) is negative?

Strictly in the case of a $1\!:\!1$ compound of $\ce{AB}$ on a mole basis, if the net change in oxidation number is negative, then you would call $\ce{AB}$ an oxidizing agent, because it pulled electrons away from something else (it oxidized that other thing) in order to achieve the net negative change in oxidation number.

For $\ce{AB}$ compounds not in a $1\!:\!1$ molar ratio, a universal principle can't be stated $-$ it depends on the particulars of the mole ratio and the individual changes in oxidation number.

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Actually it is the full reactants that are the oxidizing and reducing agents. We just specify atoms for these roles as a simplifying assumption; for example, when $\ce{KMnO4}$ acts as an oxidizing agent in acid, we call the manganese the oxidizing agent because the change in manganese oxidation state from +7 to +2 accounts for the five electrons taken up by each permanganate ion during the course of reaction. But, technically, it is the permanganate as a whole that consumes the electrons, not just the manganese.

Such a simplifying assumption is bound not to work all the time, and we may need to treat the whole compound as the oxidizing or reducing agent. See this question for an example of such a situation.

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