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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.

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    $\begingroup$ $C$ being $\kappa$-generated suffices. I mean the existen of generating (trivial) cofibrations with $\kappa$-small source and target. $\endgroup$
    – Fernando Muro
    Jul 21, 2016 at 18:38
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    $\begingroup$ As long as X is nonempty, a necessary and sufficient condition for [X,C] to be κ-generated is that C itself is κ-generated. The generating acyclic cofibrations for the projective model structure have the form R⊗j, where R is representable and j is a generating acyclic cofibration in C. The (co)domains of R⊗j are κ-small if the (co)domains of j are. $\endgroup$
    – Dmitri Pavlov
    Jul 21, 2016 at 18:40

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