The book on abstract algebra I'm going through has the following exercise. Let $A' \cong A'', B' \cong B''$ be two pairs of isomorphic sets. Further, let them be pairwise disjoint: $A' \cap B' = \emptyset$, $A'' \cap B'' = \emptyset$. Prove that $A' \cup B' \cong A'' \cup B''$ using the property of terminal objects being unique up to an isomorphism.
I understand that, due to them being disjoint, $A' \cup B'$ is the coproduct of $A', B'$, and so is $A'' \cup B''$ w.r.t. $A'', B''$. What's a corpoduct of two sets $A', B'$? It's an initial object in the category $\mathbf{Set}^{A', B'}$. Correspondingly, the coproduct of $A'', B''$ is an initial object in $\mathbf{Set}^{A'', B''}$. So, since all initial objects are isomorphic, we get… But wait, those are two different categories, even though $A' \cong A''$ and $B' \cong B''$!
How do I proceed in this case without essentially duplicating a set-theoretic proof, especially given that it's the very early chapters of the book, and I'm not supposed to know about functors or any other fancy ways to relate categories between each other?
Best Answer
Let $\mathcal{C}$ be a category with binary coproducts, $a,b,a',b'\in\text{Obj}(\mathcal{C})$, such that $a\cong a'$ and $b\cong b'$ (which means that there exist morphisms $f\colon a\to a'$, $g\colon a'\to a$, $f'\colon b\to b'$, $g'\colon b'\to b$ with identity compositions). Try to prove that a triplet $(c,s_a,s_b)$ is an initial object in $\mathcal{C}^{a,b}$ if and only if the triplet $(c,s_a\circ g,s_b\circ g')$ is an initial object in $\mathcal{C}^{a',b'}$.