I recall that $\Delta$ is the category whose objects are of the form $\textbf{n}=\{0,1,...,n\}$ and morphisms are (weakly) order preserving maps.
Let $\mathcal{C}$ be a category, and let $\mathcal{C}^{\Delta}=[\Delta, \mathcal{C}]$ be the functor category of cosimplicial objects in $\mathcal{C}$.
There is a functor $\text{ev}_0:\mathcal{C}^{\Delta} \to \mathcal{C}$ which takes a cosimplicial object $X[-]$ to its value at $0$, $X[0]$.
There is also a functor $r:\mathcal{C} \to \mathcal{C}^{\Delta}$ taking an object $C$ to the constant functor $rC$ such that $rC[n]=C$ for all $n$.
I read the claim that we have an adjunction $$\text{ev}_0 \dashv r$$ and I would like to prove it.
Given a natural transformation $\eta: X[-] \Rightarrow rC$, I can of course send it to the map $\eta_0:X[0]\to C.$
On the other hand, I can consider the diagram $$\cdots\to X[n]\to \cdots \to X[1]\to X[0]$$ wehere each $$\alpha_{n,n-1}:X[n] \to X[n-1]$$ is induced by the surjection $\textbf{n}\to \textbf{n-1}$ sending $n \mapsto n-1$ and $i \mapsto i$ for all $i<n$.
So given a map $f:X[0] \to C,$ I can inductively define $$f_0=f$$ $$f_i=f_{i-1}\alpha_{i,i-1}$$
I think that if I prove this family $\{f_i\}_i$ defines a map of cosimplicial sets, i.e. a natural transformation, I am done. But I don't know how to do that w.r.t. general maps $X[i]\to X[j].$