Two random commentsRegarding schemes: on a scheme $X$, the functor of global sections is exact iff no quasicoherent sheaves have higher cohomology. By Serre's criterion for affineness, for reasonable schemes this is equivalent to $X$ being affine, essentially because in this case global sections is a functor $\text{QCoh}(X) \to \Gamma(X, \mathcal{O}_X)\text{-Mod}$ and exactness is one of the only obstructions to it being an equivalence.
$1$ is compact projective if it's dualizable, since in that case $\text{Hom}(1, -) \cong 1^{\ast} \otimes (-)$. Maybe this was too obvious to write down. In particular most monoidal categories presented by string diagrams you might want to write down should have the property that every object is dualizable.
On a scheme $X$, the functor of global sections is exact iff no quasicoherent sheaves have higher cohomology. By Serre's criterion for affineness, for reasonable schemes this is equivalent to $X$ being affine, essentially because in this case global sections is a functor $\text{QCoh}(X) \to \Gamma(X, \mathcal{O}_X)\text{-Mod}$ and exactness is one of the only obstructions to it being an equivalence.