I wanted to ask the following question,
Suppose $\mathbf{M}$ a cofibrantly generated model category and $I,~J$ two small categories. Suppose that $F:J\rightarrow \mathbf{M}^{\mathrm{I}}$ is a functor. Is it true that the map
$\mathrm{hocolim}\_{J}(F(i))\rightarrow (\mathrm{hocolim}\_J~F)(i)$ is a weak equivalence in $\mathbf{M}$ ?


For more precision: $F(i)$ is the evaluation of the functor $F:J\times I\rightarrow \mathbf{M} $ at $i\in I$, and $\mathrm{hocolim}\_J~F$ is an object in the model category $\mathbf{M}^{I}$.
The categories $\mathbf{M}^{I}$, $\mathbf{M}^{J\times I}
$ are equipped with the projective model structure.


Thank you.



Fubini theorem for hocolim
This property holds actually for right derivable categories in the sense of:


MR2729017 Reviewed Cisinski, Denis-Charles Catégories dérivables. (French) [Derivable categories] Bull. Soc. Math. France 138 (2010), no. 3, 317–393.


At least under suitable finiteness assumptions on $I$ and $J$. It also holds for arbitrary small categories $I$ and $J$ when working on a homotopically complete right derivable category. Model categories are examples of these (cofibrant generation is not needed).


Cisinski shows that the homotopy categories of diagrams on a (homotopically complete) right derivable category $\mathcal{C}$ form a right derivator whose domain is the category of directed finite categories (or all small categories in the homotopically complete case). In particular, one of the axiom says that if $u\colon A\rightarrow B$ is any functor in the domain and $u\_!\colon\operatorname{Ho}(\mathcal{C}^A)\rightarrow \operatorname{Ho}(\mathcal{C}^B)$ is the left adjoint of the restriction along $u$ functor $u^{\*}\colon\operatorname{Ho}(\mathcal{C}^B)\rightarrow \operatorname{Ho}(\mathcal{C}^A)$ then for any $F\colon A^{\operatorname{op}}\rightarrow \mathcal{C}$ and any object $b\in B$ the formula $u\_!(F)(b)=p\_!(A\downarrow b\rightarrow A\stackrel{F}\rightarrow \mathcal{C})$ holds. Here $p\_!\colon\operatorname{Ho}(\mathcal{C}^{A\downarrow b})\rightarrow \operatorname{Ho}(\mathcal{C})$ is the usual homotopy colimit.


In your case, take $u\colon I\times J\rightarrow I$ to be the projection onto the first factor. Then $(\operatorname{hocolim}\_JF)(i)=u\_{!}(F)(i)$ by definition, the functor $J\rightarrow(I\times J)\downarrow i\colon j\mapsto (i,j)$ is cofinal, and hence $(\operatorname{hocolim}\_JF(i))=p\_!((I\times J)\downarrow i\rightarrow I\times J\stackrel{F}\rightarrow \mathcal{C})$.


