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Let $C$ be a small category, and $D$ any category, and $D^C$ the functor category of functors from $C$ to $D$.

The diagonal functor $\Delta\colon D\to D^C$ takes an object $a\in D$ to the constant functor at that object: $\Delta(a)(x) = a$ for all $x\in C.$

The limit functor $\lim\colon D^C \to D$ takes a functor $F\colon C\to D$ to its limit, $\lim F$, the initial cone over $F$.

There is an adjunction $\operatorname{Hom}_{D^C}(\Delta(a),F) = \operatorname{Hom}_{D}(a,\lim F).$ This is more or less just a restatement of the universal property of the limit.

Now if we set $D=\text{Sets},$ then $D^C=\text{Sets}^C=\hat{C}$ is the category of presheaves on $C$. $\Gamma(F)=\operatorname{Hom}_{\hat{C}}(1,F)$ is the global sections functor. In this setting we again have an adjunction with the diagonal functor: $\operatorname{Hom}_{\hat{C}}(\Delta(a),F) = \operatorname{Hom}_{\text{Sets}}(a,\Gamma(F)).$

By the uniqueness of adjoint functors, we can conclude that $\Gamma\cong\lim$, yes? Assuming that's true, that seems rather odd, can I have some context to that isomorphism to make it seem more natural or less surprising? How does it look for enriched presheaves, when $D$ is not $\text{Sets}$?

Is there some significance to the fact that the limit of a functor is just the set of natural transformations from the constant functor at the terminal object?

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Your situation is actually a special case of a general fact :

Proposition : If $G:\mathcal{X}\to \mathbf{Sets}$ is a functor that has a left adjoint $F$, then $G$ is represented by $F(1)$ (where $1$ is just a one-element set).

The proof is easy : just notice that $\operatorname{Hom}_\mathbf{Sets}(1,\_)$ is naturally isomorphic to the identity functor on $\mathbf{Sets}$, and then the adjunction gives you directly an isomorphism $$G(x)\cong \operatorname{Hom}_\mathbf{Sets}(1,G(x))\cong \operatorname{Hom}_{\mathcal{X}}(F(1),x)$$ natural in $x$.

In your case $G=\lim:\mathcal{X}\to \mathbf{Sets}$ is right adjoint to $\Delta$, thus it must be represented by $\Delta(1)$, i.e. is must be isomorphic to $\operatorname{Hom}_{[C,\mathbf{Sets}]}(\Delta(1),\_)$, which is how you've defined the global sections functor.

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  • $\begingroup$ I guess it was the choice to elide the presence of a diagonal functor in the sheaf notation. All we're really saying is, via the first adjunction, that $\operatorname{Hom}(\Delta 1,F)=\operatorname{Hom}(1,\lim F)$, and then for any set $S$, $\operatorname{Hom}(1,S)\cong S.$ This is just an obvious fact about sets that they are determined by their elements, which we might call the axiom of extensionality or the fact that Set is a cocomplete well-pointed category. I guess my question turned out kind of dumb. $\endgroup$ – ziggurism Dec 4 '19 at 16:45

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