Consider the inclusion of presheaves on $\mathbb{C}$ into families of sets indexed by $\mathbb{C}$-objects (which proceeds by forgetting the action on morphisms). Is there a left adjoint to this inclusion functor?

I thought that I had constructed something that seemed reasonable, but now I am doubtful whether it is in fact adjoint.

Consider the inclusion of presheaves on $\mathbb{C}$ into families of sets indexed by $\mathbb{C}$-objects (which proceeds by forgetting the action on morphisms). Is there a left adjoint to this inclusion functor?

I thought that I had constructed something that seemed reasonable, but now I am doubtful whether it is in fact adjoint.

EDIT: I have just realized that whilst it may not have a left adjoint (does anyone know?), it at least has a right adjoint (which is in fact the thing that I had originally constructed, hoping it was the left adjoint). Basically, take the right kan extension of a presheaf along the inclusion $\iota:\vert\mathcal{C}\vert\to \mathcal{C}^{op}$!