Continuing my recent investigations into adjunctions, I've come to understand how products are defined in terms of adjunctions. A category $\mathscr{C}$ has products if there is a right adjoint to the diagonal function $\Delta\colon \mathscr{C} \to \mathscr{C}\times\mathscr{C}$, namely $\times\colon \mathscr{C}\times\mathscr{C} \to \mathscr{C}$. The product of two objects $A$ and $B$ is $\times(A,B)$, more typically denoted $A \times B$.
Now, I know that coproducts are dual to products, and that in the direct way of describing coproducts (i.e., as they're typically presented before adjunctions have been covered), they're just presented as dual to products: reverse the arrows in the defining diagram and be done. As such, I think that we can explain coproducts using adjunctions by simply saying that $\mathscr{C}^{\mathbf{op}}$ has products, and to leave it at that. So:
- Is this correct? I'm reasonably confident that it is, but I could be missing some caveat.
- Is this the typical way to characterize coproducts using adjunctions? If not, is there a simpler or more direct to do it, while still using adjunctions?
$\mathbf{C}$ having coproducts is indeed the same thing as saying $\mathbf{C}^\circ$ has products.
Usually, when this style is used, we define coproducts as left adjoints to the diagonal. More generally, limits are right adjoints to the generalized diagonal $\mathbf{C} \to \mathbf{C}^J$, and colimits are left adjoints.