Is the unit compact in a closed monoidal (co)complete category?

Solution 1:

It's not even true for all presheaf toposes.

Let $I$ be a set. The category $\textbf{Set}^I$ is a presheaf topos, and in particular, complete and cocomplete and cartesian closed. The terminal object $1$ is not, in general, finitely presentable. Indeed, a general object $X$ in $\textbf{Set}^I$ is just an $I$-indexed family of sets, and the set of morphisms $1 \to X$ is in natural bijection with $\prod_{i \in I} X_i$.

Proposition. The functor $X \mapsto \prod_{i \in I} X_i$ preserves filtered colimits if and only if $I$ is a finite set.

Proof. The "if" direction is well known.

For the "only if" direction, observe that $X \cong \varinjlim_{J \in \mathscr{P}_\textrm{f} (I)} X_{(J)}$, where $\mathscr{P}_\textrm{f} (I)$ is the set of finite subsets of $I$ and $(X_{(J)})_i = X_i$ if $i \in J$ and $(X_{(J)})_i = \emptyset$ if $i \notin J$. If $I$ is infinite, then $\prod_{i \in I} (X_{(J)})_i$ is always empty, hence $\varinjlim_J \prod_{i \in I} (X_{(J)})_i$ is also empty. But $\prod_{i \in I} X_i$ may be non-empty. So $X \mapsto \prod_{i \in I} X_i$ does not preserve filtered colimits if $I$ is infinite.　◼