This question is somehow related to my question at https://math.stackexchange.com/questions/683915/derived-pseudo-functor and to the question here: A homotopy commutative diagram that cannot be strictified .
Consider a model category $M$. Let us assume that, for all small category $S$, we can endow the category $Fun (S, M) $ with the injective model structure.
So, considering a given small category $S$, we got the notion of a homotopy limit $ Ho(Fun (S, M))\to Ho(M) $.
My first question is: which would be the correct notion of homotopy limit $ Fun (S, Ho(M))\to Ho(M) $? The second question is: how can I study the existence of such a homotopy limit?
I know that if the diagonal functor $ \Delta : Ho (M)\to Fun (S, Ho(M)) $ has a right adjoint, then the answer is clear. But, also, I do know that this situation is rare for model categories $ M $ (considering $S$ being not discrete). However, in the general situation, I guess that the correct notion of a such homotopy limit is the following:
Consider the projections $\lambda : M\to Ho(M) $ and $ \alpha : Fun(S, M)\to Ho (Fun (S, M)) $. The functor $ Fun ( S, \lambda ) $ has a total derived functor $ i: Ho ( Fun (S, M) ) \to Fun (S , Ho (M) ) $. I would define the homotopy limit $ Fun ( S, Ho(M)) $ as being a right Kan extension of the usual homotopy limit $ Ho(Fun (S, M))\to Ho (M) $ along thar derived functor $ i $.
I guess that, one example of such a homotopy limit is the 2 (weak) - limit (homotopy limit) of a pseudo functor $ A: S\to Cat $.
But, if this is the correct notion of such a homotopy limit, my question remains: how can I study the existence of that right Kan extension? Where can I find about this kind of homotopy limit?
Thank you in advance