When the points of coproduct are the coproduct of points

Question

Let $C$ be a category with coproduct. The Yoneda functor $C^{op}\to Psh(C)$ preserves all limits but not colimits.

Suppose that $C$ has a terminal object, say $*$.

My question: can we say anything whether $\hom(*, A\sqcup B)=\hom(*, A)\sqcup \hom(*, B)$?

For a general diagram $J$ in $C$, suppose that $J$ is connected in certain sense. Under what condition, one can expect $$ \mathrm{cocone}(J, A\sqcup B)=\mathrm{cocone}(J, A)\sqcup \mathrm{cocone}(J, B). $$ (It is easy to see that this holds for $Sets$.)

Answer 1

For the purposes of this question it is best to assume that $\mathcal{C}$ has at least all finite coproducts, if not small coproducts. Let us say a connected object is an object $X$ in $\mathcal{C}$ such that $\mathcal{C} (X, -) : \mathcal{C} \to \mathbf{Set}$ preserves finite coproducts. For example, if $\mathcal{C} = \mathbf{Top}$, then this coincides with the notion of (non-empty) connected topological space. Note that the initial object of $\mathcal{C}$ is never connected.

Let $\mathcal{J}$ be a small category. It is known that the functor $\varprojlim : [\mathcal{J}, \mathbf{Set}] \to \mathbf{Set}$ preserves small coproducts if and only if $\mathcal{J}$ is connected. Thus, if $\mathcal{J}$ is a connected category, $X : \mathcal{J} \to \mathcal{C}$ is a diagram of connected objects, and ${\varinjlim}_\mathcal{J} X$ exists in $\mathcal{C}$, then ${\varinjlim}_\mathcal{J} X$ is also a connected object in $\mathcal{C}$. In other words, a connected colimit of connected objects is connected. Conversely, the coproduct of two connected objects is never connected: indeed, the canonical map $$\mathcal{C} (X + Y, X) \amalg \mathcal{C} (X + Y, Y) \to \mathcal{C} (X + Y, X + Y)$$ is injective but never surjective. (What would the preimage of $\mathrm{id}_{X + Y}$ be?)

Now suppose $\mathcal{C}$ has a terminal object $1$. There are very many examples of well-behaved categories where $1$ fails to be connected: for instance, if $\mathcal{C}$ is $\mathbf{Sh} (X)$ or $\mathbf{Top}_{/ X}$ for some topological space $X$, then $1$ is connected if and only if $X$ is. But it sometimes happens that $1$ is connected, e.g. $\mathbf{Set}$, $\mathbf{sSet}$, $\mathbf{Cat}$...