I am starting to study category theory and I have to prove that the diagonal functor is the right-adjoint of the coproduct functor.
I would like to write this using the Hom-set definition adjunction. But I am not sure what do I have to prove.
Do I have to show that
$$\hom_\mathcal{C}(Y+Y,X)\cong \hom_\mathcal{D}(Y,X\times X)$$
for all $X\in \mathcal{C}$ and $Y\in \mathcal{D}$?
Or how do you write that the diagonal functor is the right-adjoint of the coproduct functor using the Hom-set definition?
Note that the diagonal morphism is $\Delta: X\rightarrow X\times X; x\mapsto(x,x)$.
Best Answer
So, writing $X\coprod Y$ for the disjoint union, you want to prove that there is an isomorphism $$\text{hom}_\mathcal C(X\coprod Y,Z)\simeq \text{hom}_\mathcal {C\times C}((X,Y),(Z,Z))$$(and these isomorphisms need to be natural). If you read it from right to left, you want to show there is an isomorphism between morphisms $(X,Y)\rightarrow(Z,Z)$ in $\mathcal C\times\mathcal C$, that is pairs $(f:X\rightarrow Z,g:Y\rightarrow Z)$, and morphisms $X\coprod Y\rightarrow Z$, then you can see that this isomorphism comes directly from the universal property of coproducts. The inverse morphism is given by precomposing $X\coprod Y\rightarrow Z$ with the inclusions $X\rightarrow X\coprod Y$ and $Y\rightarrow X\coprod Y$ (again, this is the inverse because of the universal property), this is the isomorphism you're looking for. Then you just need to prove that these isomorphisms for every $X,Y,Z\in Ob(\mathcal C)$ form a natural transformation between the functors $\mathcal{(C\times C)^{op}\times C}\rightarrow\mathbf{Set}$ $$((X,Y),Z)\mapsto \text{hom}_\mathcal C(X\coprod Y,Z)\text{ and }((X,Y),Z)\mapsto \text{hom}_{\mathcal C\times \mathcal C}((X,Y),(Z,Z))$$this shouldn't be too hard to see (maybe a little tedious).
I want to point out that this is a special case of a more general result: Given two categories $\mathcal C,\mathcal D$, let $\mathcal C^\mathcal D$ be the functor category, then we have a diagonal functor $\Delta:\mathcal C\rightarrow \mathcal C^\mathcal D$ sending each object $X\in Ob(\mathcal C)$ to the functor $\mathcal D\rightarrow\mathcal C$ with constant value $X$ on objects and identity of $X$ on morphisms. You can prove that $\Delta$ is right adjoint to the functor (assuming it exists) $\mathcal C^\mathcal D\rightarrow\mathcal C$ sending each functor $F:\mathcal D\rightarrow\mathcal C$ to its colimit. You can also prove that $\Delta$ is left adjoint to the functor sending each $F:\mathcal D\rightarrow \mathcal C$ to its limit. In your case, we're taking as $\mathcal D$ the category $\{\bullet,\bullet\}$ with two objects and only identity morphisms.