Let $\mathcal C$ be a category with a zero-object and $(A,B)$ be a pair of objects having a coproduct $(A\oplus B, i_A : A \to A \oplus B, i_B : B \to A \oplus B)$. We obtain unique morphisms $r_A : A \oplus B \to A$ and $r_B : A \oplus B \to B$ such that $r_A i_A = id, r_A i_B = 0$ and $r_B i_B = id, r_B i_A = 0$.
Let us write $(A\oplus B, i_A, i_B)^* = (A \oplus B, r_A, r_B)$ and call it the dual associate of $(A\oplus B, i_A,, i_B)$.
Example 1. Let $\mathcal C$ be the category of abelian groups and group homomorphims. Then $(A\oplus B, i_A, i_B)^*$ is the product of $A, B$. It is of course characterized by a universal property.
Example 2. Let $\mathcal C$ be the category of pointed topological spaces and basepoint-preserving continuous maps. Then $(A \oplus B, i_A, i_B)$ is the wedge $A \vee B$ with canonical embeddings $i_A, i_B$. Hence $(A \vee B, i_A, i_B)^* = (A \vee B, r_A, r_B)$ with canonical retractions $r_A : A \vee B \to A$ and $r_B : A \vee B \to B$. I am not aware of any universal property characterizing this system.
Questions:
Is there a characterizing universal property in example 2?
More, generally, under what conditions on $\mathcal C$ can the dual associate be characterized by some universal property?