I wanted to ask the following question, Suppose $\mathbf{M}$ a cof 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.

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.