Suppose we have:

-) A $2$-category $\mathsf{J}$

-) A natural transformation of $\alpha : M \Longrightarrow W : \mathsf{J} \longrightarrow \mathsf{Cat}$

-) A functor $X:\mathsf{J} \longrightarrow \mathsf{Cat}$

Has anyone written down conditions such that the functor:

$$ \alpha^*:[\mathsf{J},\mathsf{Cat}](W,X) \longrightarrow [\mathsf{J},\mathsf{Cat}](M,X) $$

admits a left or right adjoint functor?

I assume the conditions will be the existence, for each $j\in\mathsf{J}$, of left/right Kan extensions of

$$ M_j \longrightarrow X_j $$

along

$$ M_j \longrightarrow X_j $$

together with preservation of left/right Kan extensions by the functors $W_f:W_i \longrightarrow W_j$ and something similar about $M$ and $W$ and $\alpha$.

This feels like something that someone may have written down.

Suppose we have:

-) A $2$-category $\mathsf{J}$

-) A natural transformation of $\alpha : M \Longrightarrow W : \mathsf{J} \longrightarrow \mathsf{Cat}$

-) A functor $X:\mathsf{J} \longrightarrow \mathsf{Cat}$

Has anyone written down conditions such that the functor:

$$ \alpha^*:[\mathsf{J},\mathsf{Cat}](W,X) \longrightarrow [\mathsf{J},\mathsf{Cat}](M,X) $$

admits a left or right adjoint functor?

I assume the conditions will be the existence, for each $j\in\mathsf{J}$, of left/right Kan extensions of

$$ M_j \longrightarrow X_j $$

along

$$ M_j \longrightarrow X_j $$

together with preservation of left/right Kan extensions by the functors $X_f:X_i \longrightarrow X_j$ and something similar, but about pre-composition, about $M$ and $W$ and $\alpha$.

This feels like something that someone may have written down.