Let $\mathcal{C}$ be a small finitely complete category equipped with a Grothendieck (pre)topology $\tau$. For $\ast$ the terminal category (one object, one morphism), denote by $i: \ast \to \mathcal{C}$ the inclusion of the terminal object into $\mathcal{C}$. Equip $\ast$ with the trivial topology in which the only covering is the identity morphism of the unique object onto itself.


By the usual yoga of functoriality of presheaves, $i$ induces a triple of adjoint functors $i\_! \dashv i^\ast \dashv i\_\ast$ between the categories of presheaves on $\mathcal{C}$ and $\ast$. Trivially $i$ preserves coverings and is left exact —that is, it furnishes a morphism of sites. Then $i^\ast$ takes sheaves to sheaves and so the pair $i\_! \dashv i^\ast$ descends to an adjoint pair, $a i\_! \dashv i^\*$ ($a$ denoting sheafification), of functors between the appropriate categories of sheaves, which is in fact nothing but the geometric morphism of global sections of $\mathrm{Sh}(\mathcal{C}, \tau)$ consisting of the global sections functor $\Gamma: \mathsf{Sh}(\mathcal{C}, \tau) \to \mathsf{Set}$ and its left adjoint, $\mathrm{const}: \mathsf{Set} \to \mathsf{Sh}(\mathcal{C}, \tau)$, taking a set $A$ to the locally constant sheaf on $A$.


It seems to me that $i$ is also a cocontinuous functor (in the terminology of SGA4 or the [Stacks Project, section 19](http://stacks.math.columbia.edu/download/sites.pdf); other texts, such as MacLane-Moerdijk [VII.10.Theorem 5] call it a functor having the covering lifting property), making the above geometric morphism of global sections into a [local geometric morphism](http://ncatlab.org/nlab/show/local+geometric+morphism). The latter means that $i\_\ast$ takes sheaves to sheaves ([Stacks Project, Lemma 19.2](http://stacks.math.columbia.edu/download/sites.pdf)), so that there is a triple of adjoint functors $a i\_! \dashv i^\ast \dashv i\_\ast$ between the categories of sheaves on $(\mathcal{C}, \tau)$ and $\ast$. However, an explicit calculation (see [Stacks Project, section 18](http://stacks.math.columbia.edu/download/sites.pdf)) of $i\_\ast$ yields that $i\_\ast(A)$ is the constant presheaf on $A$, which is not in general a sheaf.


I am clearly missing something, but I cannot fathom what it is. Can anybody?



Cocontinuous functor out of the terminal category
The functor $i$ does *not* have the cover lifting property in general. If it did, then every epimorphism $X \to 1$ in $\mathbf{Sh}(\mathcal{C}, \tau)$ would be an isomorphism, or equivalently, every $\tau$-cover of $1$ in $\mathcal{C}$ must contain a split epimorphism. (Check the definition of "cover lifting property" carefully!)


