Introduction:
A double displacement reaction, also known as a double replacement reaction or metathesis reaction, is a chemical reaction where the positive ions of two different compounds exchange places with each other to form two new compounds. This type of reaction typically occurs in aqueous solutions where the reactants are ionic compounds.

Explanation:
A double displacement reaction can occur even when the products are highly soluble or highly ionized. The solubility or ionization of the products does not prevent the reaction from taking place. However, the extent and rate of the reaction may be influenced by the solubility or ionization of the products.

Factors affecting the reaction:
1. Solubility: Highly soluble products will remain as dissociated ions in the solution, which increases the likelihood of the double displacement reaction taking place. If the products are insoluble, they may precipitate out of the solution, reducing the chances of a complete reaction.

2. Ionization: Highly ionized products will exist predominantly as ions in the solution. This enhances the chances of ion exchange occurring during the double displacement reaction. However, even if the products are highly ionized, the reaction can still proceed as long as the ions are free to move and interact with each other.

Example:
Let's consider the reaction between silver nitrate (AgNO3) and sodium chloride (NaCl) in aqueous solution:

AgNO3 + NaCl → AgCl + NaNO3

In this reaction, both silver chloride (AgCl) and sodium nitrate (NaNO3) are highly soluble in water and highly ionized. Despite the high solubility and ionization of the products, the double displacement reaction still occurs. The silver ion (Ag+) from silver nitrate exchanges with the sodium ion (Na+) from sodium chloride to form silver chloride, which precipitates out of the solution. The nitrate ion (NO3-) from silver nitrate combines with the sodium ion to form sodium nitrate.

Conclusion:
In conclusion, a double displacement reaction can take place even when the products are highly soluble or highly ionized. The solubility and ionization of the products may affect the rate and extent of the reaction but do not prevent the reaction from occurring. It is important to consider the nature of the reactants and the conditions under which the reaction takes place to determine the likelihood and outcome of a double displacement reaction.