Suppose I have a Quillen pair $F \dashv G$ with $F:\text{Psh}(\mathcal{C}\times{\Delta}) \to \mathcal{M},$ and consider also the category of cosimplicial objects in $\mathcal{M}$ denoted $\mathcal{M}^{\Delta}$ where we have the Reedy model structure. Let $h_{(C,n)}$ be a representable functor in $\text{Psh}(\mathcal{C}\times{\Delta})$.
How do I prove that the cosimplicial object $\Gamma C:n \mapsto Fh_{(C,n)}$ is cofibrant in $\mathcal{M}^{\Delta}$?
It would suffice to prove that $L_nFh_{(C,n)}\to Fh_{(C,n)}$ is a cofibration in $\mathcal{M}$ for every $n.$
But how do I compute the latching space $$L_nFh_{(C,n)}=\text{colim}_{m \to n}Fh_{(C,m)}=F(\text{colim}_{m\to n}h_{(C,m)})?$$
where the index of the colimit I recall is the category of maps $\textbf{m} \to \textbf{n}$ for $m<n$.
Maybe $$\text{colim}_{m\to n}h_{(C,n)}=\text{colim}_{m\to n}(h_{C}\times h_n)=h_C \times \text{colim}_{m\to n}h_n=h_C\times \partial\Delta^n$$ and $$h_C \times \partial \Delta^n \to h_C \times h_n$$ is a cofibration?
As I noted in the comments, given the work in the question, it suffices to prove that the functor $\newcommand\op{{\text{op}}}\newcommand\C{\mathcal{C}}h_C\times - : \newcommand\sSet{\mathbf{sSet}}\sSet\to [\C^\op,\sSet]$ is left Quillen. In turn, it suffices to prove that this functor is left adjoint to $\newcommand\ev{\operatorname{ev}}\ev_C : [\C^\op,\sSet]\to\sSet$, since this functor is right Quillen by definition of the projective model structure.
You've asked in the comments about how to prove that these functors are in fact adjoint, so that's what I will address in this answer. I'm going to use the convention that lowercase $c$s are objects in the category $\C$.
I think this is easier using ends to unpack and repack natural transformations. A quite extensive reference on the topic of (co)ends is Fosco Loregian's Co/end calculus. $$ \newcommand\Set{\mathbf{Set}} \begin{align*} [\C^\op,\sSet](h_c\times K,F) &\simeq \int_{c'\in\C^\op}\sSet(h_c(c')\times K,Fc') \\ &\simeq \int_{c'\in\C^\op}\int_{n\in\Delta^\op} \Set(h_c(c')\times K_n,F(c',n)) \\ &\simeq \int_{n\in\Delta^\op} \int_{c'\in\C^\op} \Set(K_n,\Set(h_c(c'),F(c',n))) \\ &\simeq \int_{n\in\Delta^\op} \Set\left(K_n,\int_{c'\in\C^\op}\Set(h_c(c'),F(c',n))\right) \\ &\simeq \int_{n\in\Delta^\op} \Set\left(K_n,F(c,n)\right) \\ &\simeq \sSet\left(K,Fc\right) \\ \end{align*} $$
However, a direct proof may also be given, at the cost of essentially reproving the Yoneda lemma.
Given a natural transformation $\alpha : h_c\times K\to F$, we can consider the component $\alpha_c : h_c(c)\times K\to Fc$, and then restrict this to $1_c\times K \cong K \to Fc$, producing a map of simplicial sets from $K$ to $Fc$.
Conversely, given $a : K\to Fc$, for each $f:c'\to c$, define $\alpha_f : K\to Fc'$ by $\alpha_f = f^* \circ a$. Then define $\alpha_{c'} : h_c(c')\times K \to Fc'$ to be $\alpha_f$ on the component $f\times K$. You can check that this defines a natural transformation $\alpha : h_c\times K \to F$, and that these procedures give a bijection between natural transformations $h_c\times K \to F$ and maps of simplicial sets $K\to Fc$, as desired.