# projective model structure on presheaves , hom-functors are always cofibrant

Why hom-functors are always cofibrant in the projective model structure in $[\cal T,\cal V]$? The claim is here on page 5.

• This is probably using the assumption that all objects of $V$ are cofibrant. – Kevin Carlson Jan 23 '16 at 22:29
• But natural transformations in $[\cal T,\cal V]$ are cofibrant if the components i.e. arrows in $\cal V$ are cofibrations,not objects.Moreover, projective model structure speak about fibrations,and not cofibrations. – user122424 Jan 24 '16 at 16:58
• Cofibrant is an adjective that applies to objects, I have no idea what you mean by a cofibrant natural transformation. – Kevin Carlson Jan 24 '16 at 21:52