Apologies if this is a stupid question: Let $C$ be a cofibrantly generated model category ($\mathbf{Edit}$: Combinatorial) and let $[X,C]$ be functor category equipped with the projective model structure.

Recall that for an infinite cardinal $\kappa$, a model structure on a category $D$ is $\kappa$-generated if every (trivial) cofibration in $D$ is a transfinite composition of pushouts of coproducts of (trivial) cofibrations between $\kappa$-small objects.

Are there any conditions on $C$ (or something else) that would ensure that the model structure on $[X,C]$ is $\kappa$-generated?

It seems like a bit of a pain to manually check if the $\kappa$-generated condition holds.