Let $C$ be a category. Consider $Psh(C)$ the category of presheaves on $C$. It is a well known fact that the evaluation functor at $c$ an object of $C$ of presheaves preserves both limits and colimits yet I can't find in the littérature nor compute right and left adjoints. If someone knows a formula for both adjoints it would be very helpful.
Category Theory – Understanding Adjoints to Evaluation Functors
adjoint-functorscategory-theory
Best Answer
The left adjoint $L_c$ to evaluation-at-$c$ is very simple; left adjoints preserve colimits and every set is a coproduct of $1$ with itself. Consequently,
$$ L_c(X) = X \cdot L_c(1)$$
where $\cdot$ means to take the $X$-fold coproduct of $L_c(1)$ with itself. Finally,
$$ \mathbf{PSh}(\mathcal{C})(L_c(1), F) \cong \mathbf{Set}(1, F(c)) \cong F(c) $$
therefore, $L_c(1)$ is the functor $\mathcal{C}(-, c)$ represented by $c$. That is,
$$ L_c(X) = X \cdot \mathcal{C}(-, c) $$
The right adjoint $R_c$ is even simpler:
$$ R_c(X)(d) \cong \mathbf{PSh}(\mathcal{C})(\mathcal{C}(-, d), R_c(X)) \cong \mathbf{Set}(\mathcal{C}(c, d), X) \cong X^{\mathcal{C}(c,d)}$$
That is,
$$ R_c(X) \cong X^{\mathcal{C}(c, -)}$$
(thanks to Andreas Blass for reminding me of the argument)
The evaluation-at-c functor $F \to F(c)$ is, incidentally, given by the functor
$$ \mathbf{Set}^{\mathcal{C}^\circ} \to \mathbf{Set}^1 $$
induced by the inclusion $1 \to \mathcal{C}^{\circ}$ that identifies $c$, so both adjoints are special cases of the fact this functor has adjoints. I don't remember how this works out off the top of my head, but it should probably be easier to find a reference for.